Combinatorial Modulus on Boundary of Right-Angled Hyperbolic Buildings
CCombinatorial Modulus on Boundary of Right-AngledHyperbolic Buildings
Antoine ClaisLaboratoire Paul PainlevéUniversité Lille 159655 Villeneuve d’Ascq, France [email protected]
October 15, 2018
Abstract
In this article, we discuss the quasiconformal structure of boundaries of right-angledhyperbolic buildings using combinatorial tools. In particular we exhibit some examplesof buildings of dimension 3 and 4 whose boundaries satisfy the combinatorial Loewnerproperty. This property is a weak version of the Loewner property. This is motivatedby the fact that the quasiconformal structure of the boundary led to many results ofrigidity in hyperbolic spaces since G.D. Mostow. In the case of buildings of dimension2, many work have been done by M. Bourdon and H. Pajot. In particular, the Loewnerproperty on the boundary permitted them to prove the quasi-isometry rigidity of right-angled Fuchsian buildings.
Keywords:
Boundary of hyperbolic space, building, combinatorial modulus, combinatorialLoewner property, quasi-conformal analysis.
The origin of the theory of modulus of curves in compact metric spaces must be found inthe classical theory of quasiconformal maps in Euclidean spaces (see [Väi71] or [Vuo88]).Quasiconformal maps are maps between homeomorphisms and bi-Lipschitz maps. The aimof the classical theory is to describe the regularity of quasiconformal maps in R d and toexhibit invariants under these maps. The notion of abstract Loewner space, introduced byJ. Heinonen and P. Koskela (see [HK98] or [Hei01]), intends to describe metric measuredspaces whose quasiconformal maps have a behavior of Euclidean flavor.1 a r X i v : . [ m a t h . G R ] O c t oreover, since G.D. Mostow it is known that the quasiconformal structure of theboundary of a hyperbolic space controls the geometry of the space. It turns out that thisidea extends to the case of Gromov hyperbolic spaces and groups. Finding a Loewner spaceas the visual boundary of a Gromov hyperbolic group has been useful to establish rigidityresults about the group (see [Haï09b] for a survey on those results). The idea that onewants to use to prove rigidity is that the quasi-isometries of an hyperbolic space are givenby the quasisymmetric homeomorphisms of the boundary. The Loewner property makesit possible because the classes of quasi-Moebius, quasisymmetric and quasiconformal mapsare equal in a Loewner space.It is difficulty to prove that the boundary of a hyperbolic space is a Loewner space.To do so, one needs to find a measure on the boundary that is optimal for the conformaldimension . This quasiconformal invariant has been introduced by P. Pansu in [Pan89].Finding a measure that realizes the conformal dimension and even computing the conformaldimension, are very difficult questions that we can solve in few examples for the moment.An interesting example of this kind is the work done by M. Bourdon and H. Pajot inFuchsian buildings. They proved that the boundary of these buildings are Loewner spacesand then used this structure to prove the quasi-isometry rigidity of these buildings (see[BP00]).Buildings are singular spaces introduced by J. Tits to study exceptional Lie groups.Currently buildings became a topic of interest by themselves. Among them, right-angledbuildings have been classified by F. Haglund and F. Paulin in [HP03]. They are equippedwith a wall structure and with a simply transitive group action on the chambers thatmake them very regular objects. Fuchsian buildings are right-angled hyperbolic buildingsof dimension . In light of the results by M. Bourdon and H. Pajot in dimension , we havethe questions: Question 1.1.
Are higher dimensional right-angled hyperbolic buildings rigid? What arethe quasiconformal properties of their boundaries?
The geometry of higher dimensional right-angled buildings is very close to the geometryof Fuchsian buildings. This gives hope that these questions may have interesting answers.However, the methods used in Fuchsian buildings are very specific to the dimension 2. Thusthese question are not easy. In this article, we will use combinatorial modulus for a firstapproach of the conformal structure of higher dimensional right-angled buildings.A major rigidity question related to the quasiconformal structure on the boundary isthe following conjecture due to J.W. Cannon.
Conjecture 1.2 ([CS98, Conjecture 5.1.]) . If Γ is a hyperbolic group and ∂ Γ is homeomor-phic to S , then Γ acts geometrically on H . In particular, this conjecture implies Thurston’s hyperbolization conjecture of 3-manifolds.Although Thurston’s conjecture is now a theorem by G. Perelman, Cannon’s conjecture re-mains very interesting as it is logically independent of Thurston’s conjecture.2he combinatorial modulus have been introduced by J.W. Cannon in [Can94] and byM. Bonk and B. Kleiner in [BK02] during the investigation of the quasiconformal structureof the 2-spheres to approach the conjecture and by P. Pansu in a more general context in[Pan89]. Combinatorial modulus gave birth to a weak version of the Loewner property: the
Combinatorial Loewner Property (CLP). One of the feature of these modulus is that theycan be used to characterize the conformal dimension as a critical exponent on the boundary.Recently M. Bourdon and B. Kleiner (see [BK13]) gave examples of boundaries of Cox-eter groups that satisfy the CLP but that are not known for satisfying the Loewner property.They used this property to give a new proof of Cannon’s conjecture for Coxeter groups.Some of the methods they used for Coxeter groups can be adapted to the case of right-angledbuildings. This was a motivation to investigate higher dimensional right-angled buildingsusing combinatorial modulus.
In this article, we use the combinatorial modulus to investigate the quasiconformal boundaryof right-angled hyperbolic buildings. Thanks to methods in [BK13], we obtain a control ofthe combinatorial modulus on the boundary in terms of the curves contained in paraboliclimit sets (see Section 6). Then we introduce a weighted modulus on the boundary of theapartments. This allows us to control the modulus in the building by a modulus in theapartment (see Section 8). For well chosen examples, the boundary of the apartment hasa lot of symmetries that provide a strong control of the modulus. In particular, we exhibitsome examples of hyperbolic buildings in dimension 3 and 4 whose boundaries satisfy theCLP.
Theorem 1.3 (Corollary 10.3) . Let D be the right-angled dodecahedron in H or the right-angled 120-cell in H . Let W D be the hyperbolic reflection group generated by reflectionsabout the faces of D . Let ∆ be the right-angled building of constant thickness q ≥ and ofCoxeter group W D . Then ∂ ∆ satisfies the CLP. Along with this result we also give in Theorem 9.1, a characterization of the conformaldimension of the building using a critical exponent computed in an apartment.
In Section 2, we introduce the combinatorial modulus of curves in the general setting ofcompact metric spaces. Then in Section 3, we restrict to the case of boundaries of hyperbolicspaces.After these reminders, we give the main steps and ideas of the proof of Theorem 10.1in Section 4. This section is essentially a summary of the article.Then, in Section 5, we describe the geometry of locally finite right-angled hyperbolicbuildings. 3he key notion of parabolic limit sets is introduced in Section 6 where we study themodulus of curves in parabolic limit sets. This section is based on ideas used in Coxetergroups in [BK13, Sections 5 and 6]. In particular, Theorem 6.12 is the first major steptowards the proof of Theorem 10.1. As a consequence of this theorem, we obtain a firstapplication to the CLP (Theorem 6.13).In Section 7, we describe the combinatorial metric on the boundary of the group interms of the geometry of the building. This metric is useful in computing the combinatorialmodulus. Then, in Section 8 we discuss how the modulus in the boundary of an apartmentmay be related to a modulus in the boundary of the building. In particular, Theorem 8.9is the second major step necessary to prove Theorem 10.1. We use this theorem to proveTheorem 9.1 which relates the conformal dimension of the boundary of the building to acritical exponent computed in the boundary of an apartment.In Section 9, we add the constant thickness assumption for the building under whichthe results of the preceding section can be made more precise. In particular, we find thatthe conformal dimension of the boundary of the building is equal to a critical exponentcomputed in the boundary of an apartment (see Theorem 9.1). Finally in Section 10, wegather these tools to obtain examples of right-angled-buildings of dimension 3 and 4 whoseboundary satisfies the CLP (see Corollary 10.3).
Throughout this paper, we will use the following conventions. The identity element in agroup will always be designated by e . For a set E , the cardinality of E is designated by E . A proper subset F of E is a subset F (cid:32) E .If G is a graph then G (0) is the set of vertices of G and G (1) is the set of edges of G . For v, w ∈ G (0) , we write v ∼ w if there exists an edge in G whose extremities are v and w .If V ⊂ G (0) , the full subgraph generated by V is the graph G V such that G (0) V = V and anedge lies between two vertices v , w if and only if there exists an edge between v and w in G . A full subgraph is called a circuit if it is a cyclic graph C n for n ≥ . A graph is calleda complete graph if for any pair of distinct vertices v , w there exists an edge between v and w . A curve in a compact metric space ( Z, d ) is a continuous map η : [0 , −→ Z . Usu-ally, we identify a curve with its image. If η is a curve in Z , then U (cid:15) ( η ) denotes the (cid:15) -neighborhood of η for the C -topology. This means that a curve η (cid:48) ∈ U (cid:15) ( η ) if and onlyif there exists s : t ∈ [0 , −→ [0 , a parametrization of η such that for any t ∈ [0 , onehas d ( η ( s ( t )) , η (cid:48) ( t )) < (cid:15) .In a metric space Z , if A ⊂ Z then N r ( A ) is the r -neighborhood of A . The closure of A is designated by A and the interior of A by Int( A ) . If B = B ( x, R ) is an open ball and λ ∈ R then λB is the ball of radius λR and of center x . A ball of radius R is called an R -ball . The closed ball of center x and radius R is designated by B ( x, R ) .A geodesic line (resp. ray ) in a metric space ( Z, d ) is an isometry from ( R , | · − · | ) ([0 , + ∞ ) , | · − · | ) into ( Z, d ) . The real hyperbolic space (resp. Euclidean space) ofdimension d is denoted H d (resp. E d ). This article is part of my Ph.D Thesis realized at Université Lille 1 under the direction ofMarc Bourdon. I am very thankful to Marc for his support during these years. I thankFréderic Haglund for the interest he demonstrated to this work and his fruitful comments.I am also grateful to the reviewers of my Ph.D thesis Mario Bonk and Pierre-EmmanuelCaprace for their attentive reading. Eventually, I would like to thank María-Dolores ParrillaAyuso who has the ability to turn mathematics into 3d pictures.
Contents ( W, S ) . . . . . . . . . . . . . . . . . . . . . . . . . . 245.4 Buildings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.5 Graph products and right-angled buildings . . . . . . . . . . . . . . . . . . . 265.6 The Davis complex associated with Γ . . . . . . . . . . . . . . . . . . . . . . 275.7 Building-walls and residues in the Davis complex . . . . . . . . . . . . . . . 275.8 Geometric characterization of parabolic subgroups . . . . . . . . . . . . . . 305.9 Σ as a metric space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.10 Boundary of the building . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ∂ Γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.2 Modulus of curves in connected parabolic limit set . . . . . . . . . . . . . . 366.3 Application to Fuchsian buildings . . . . . . . . . . . . . . . . . . . . . . . . 41 Σ . . . . . . . . . . . . . . . . . . . . . . . . . . 447.2 Shadows on ∂ Γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477.3 Combinatorial metric on ∂ Γ . . . . . . . . . . . . . . . . . . . . . . . . . . . 517.4 Approximation of ∂ Γ with shadows . . . . . . . . . . . . . . . . . . . . . . . 54 ∂A and in ∂ Γ . . . . . . . . . . . . . . . . . . 568.2 Choice of approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578.3 Weighted modulus in ∂A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588.4 Modulus in ∂ Γ compared with weighted modulus in ∂A . . . . . . . . . . . 618.5 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 The combinatorial modulus are tools that have been developed to compute modulus ofcurves in a metric space without a natural measure. The idea is to approximate the metricspace with a sequence of finer and finer approximations. Then with these approximationswe can construct discrete measures and compute combinatorial modulus. Finally, for wellchosen examples we can check that this sequence of modulus has a good asymptotic behav-ior. In this first section, we present the general theory of combinatorial modulus in compactmetric spaces. We also recall basic definitions and facts about abstract Loewner spaces as6hey inspired the theory of combinatorial modulus. Most of this section can be found in[BK13, Section 2] to which we refer for details.In this section ( Z, d ) denotes a compact metric space. For k ≥ and κ > , a κ -approximation of Z on scale k is a finite covering G k by opensubsets such that for any v ∈ G k there exists z v ∈ v satisfying the following properties: • B ( z v , κ − − k ) ⊂ v ⊂ B ( z v , κ − k ) , • ∀ v, w ∈ G k with v (cid:54) = w one has B ( z v , κ − − k ) ∩ B ( z w , κ − − k ) = ∅ .A sequence { G k } k ≥ is called a κ -approximation of Z . Example 2.1.
For k ≥ , a − k -separated subset of Z is a subset E such that d ( z, z (cid:48) ) ≥ − k for any z (cid:54) = z (cid:48) ∈ E k . Since Z is compact any − k -separated subset of Z is finite. Let E k be a − k -separated subset of Z of maximal cardinality. Then E k satisfies the following property:for any x ∈ Z , there exists z ∈ E k such that d ( x, z ) ≤ − k .The set { B ( z, − k ) } z ∈ E k defines a -approximation at scale k of Z . Now we fix the approximation { G k } k ≥ . We construct a discrete measure based on each G k for k ≥ . Let ρ : G k −→ [0 , + ∞ ) be a positive function and γ be a curve in Z . The ρ -length of γ is L ρ ( γ ) = (cid:88) γ ∩ v (cid:54) = ∅ ρ ( v ) . For p ≥ , the p -mass of ρ is M p ( ρ ) = (cid:88) v ∈ G k ρ ( v ) p . Until the end of this subsection p ≥ is fixed. Let F be a non-empty set of curves in Z .We say that the function ρ is F -admissible if L ρ ( γ ) ≥ for any curve γ ∈ F . Definition 2.2.
The G k -combinatorial p -modulus of F is Mod p ( F , G k ) = inf { M p ( ρ ) } where the infimum is taken over the set of F -admissible functions and with the convention Mod p ( ∅ , G k ) = 0 . Mod p ( F , G k ) = inf ρ M p ( ρ ) L ρ ( F ) p , where the infimum is taken over the set of positive functions on G k and with L ρ ( F ) =inf γ ∈F L ρ ( γ ) .The next proposition allows us to see the G k -combinatorial p -modulus as a weak outermeasure on the set of curves of Z . Usually, for an outer measure the subadditivity musthold over countable sets. This is useful to get intuition on these tools. Proposition 2.3 ([BK13, Proposition 2.1.]) .
1. Let F be a set of curves and F (cid:48) ⊂ F . Then Mod p ( F (cid:48) , G k ) ≤ Mod p ( F , G k ) .2. Let F , . . . , F n be families of curves. Then Mod p ( n (cid:83) i =1 F i , G k ) ≤ n (cid:80) i =1 Mod p ( F i , G k ) . A function ρ : G k −→ [0 , + ∞ ) is called a minimal function for a set of curves F if Mod p ( F , G k ) = M p ( ρ ) . Since we only compute finite sums, minimal functions alwaysexist. Combining with a convexity argument, this also provides an elementary control ofthe modulus as follows. For F a non-empty set of curves in Z and k ≥ G k ) p − ≤ Mod p ( F , G k ) ≤ G k . In the rest of this article we mainly discuss the curves of Z of diameter larger than afixed constant. For these curves the following basic property is useful. Proposition 2.4.
Let F be a non-empty set of curves in Z . Assume that there exists d > such that diam γ ≥ d for any γ ∈ F . Then for any (cid:15) > , there exists k ≥ such that forany k ≥ k , there exists an admissible function ρ : G k −→ [0 , + ∞ ) such that ρ ( v ) ≤ (cid:15) forany v ∈ G k .Proof. Let γ ∈ F . We recall that κ denotes the multiplicative constant of the approximation { G k } k ≥ . For k ≥ log ( κ/d )log 2 , as diam γ > d the following inequality holds { v ∈ G k : v ∩ γ (cid:54) = ∅} ≥ dκ − k . Hence the constant function ρ : v ∈ G k −→ κd − k ∈ [0 , + ∞ ) is F -admissible. This finishesthe proof.A metric space Z is called doubling if there exists a uniform constant N such thateach ball B of radius r is covered by N balls of radius r/ . In doubling spaces, the G k -combinatorial p -modulus does not depend, up to a multiplicative constant, on the choiceof the approximation. 8 roposition 2.5 ([BK13, Proposition 2.2.]) . Let ( Z, d ) be a compact doubling metric space.For each p ≥ , if G k and G (cid:48) k are respectively κ and κ (cid:48) -approximations, there exists D = D ( κ, κ (cid:48) ) such that for any k ≥ D − · Mod p ( F , G k ) ≤ Mod p ( F , G (cid:48) k ) ≤ D · Mod p ( F , G k ) . Usually, we work with p ≥ fixed and with approximately self-similar spaces (seeSection 3). As these spaces are doubling, now we refer to the combinatorial modulus onscale k , omitting p and the approximation. In this subsection, we assume that ( Z, d ) is a compact arcwise connected doubling metricspace. Let κ > and let { G k } k ≥ denote a κ -approximation of Z . Moreover we fix p ≥ .A compact and connected subset A ⊂ Z is called a continuum . Moreover, if A containsmore than one point, A is called a non-degenerate continuum. The relative distance betweentwo disjoint non-degenerate continua A, B ⊂ Z is ∆( A, B ) = dist(
A, B )min { diam A, diam B } . If A and B are two such continua, F ( A, B ) denotes the set of curves in Z joining A and B and we write Mod p ( A, B, G k ) := Mod p ( F ( A, B ) , G k ) . Definition 2.6.
Let p > . We say that Z satisfies the Combinatorial p -Loewner Property (CLP) if there exist two increasing functions φ and ψ on (0 , + ∞ ) with lim t → ψ ( t ) = 0 ,such thati) for any pair of disjoint non-degenerate continua A and B in Z and for all k ≥ with − k ≤ min { diam A, diam B } one has: φ (∆( A, B ) − ) ≤ Mod p ( A, B, G k ) , ii) for any pair of open balls B , B in Z , with same center and B ⊂ B , and for all k ≥ with − k ≤ diam B one has: Mod p ( B , Z \ B , G k ) ≤ ψ (∆( B , Z \ B ) − ) . As we assume that Z is doubling, thanks to Proposition 2.5, the CLP is independent ofthe choice of the approximation. As we noticed, the modulus on scale k is an outer measure(in a weak sense) over the set of curves in Z . With the previous remarks we can interpretintuitively the two inequalities of the definition as follows: i) there are plenty of curves joining two continua,ii) the amount of curves joining two continua is a decreasing function ofthe relative distance. We present examples and properties about the CLP in Subsection 2.4.9 .3 Loewner spaces
Now we define the notion of Loewner space. This notion introduced in [HK98] has inspiredthe definition of the CLP. Moreover, the proof of many basic properties of combinatorialmodulus are directly inspired by the classical theory of modulus (see [BK13]).Now we consider ( X, d, µ ) a metric measured space. For simplicity, we assume that X is compact and that ( X, d, µ ) is a Q -Ahlfors-regular space ( Q -AR or AR) for Q > . Thismeans that there exists a constant C > such that for any < R ≤ diam X and any R -ball B ⊂ X one has C − · R Q ≤ µ ( B ) ≤ C · R Q . Note that under this assumption the measure µ is comparable to the Hausdorff measure H d .Let F be a set of curves in X . A measurable function f : X −→ [0 , + ∞ ( is said to be F -admissible if for any rectifiable curve γ ∈ F (cid:90) γ ( t ) f ( γ ( t )) dt ≥ . Note that the notion of admissibility does not use the measure on X but only the metricspace structure. Definition 2.7.
The Q -modulus of F is Mod Q ( F ) = inf (cid:110) (cid:90) X f Q dµ (cid:111) where the infimum is taken over the set of F -admissible functions and with the conventionthat Mod Q ( F ) = 0 if F does not contain rectifiable curves. As before, if A and B are two disjoint non-degenerate continua, F ( A, B ) denotes theset of curves in X joining A and B . Moreover, we write Mod Q ( A, B ) := Mod Q ( F ( A, B )) .In the literature on quasiconformal maps the pair ( A, B ) is called a condenser and themodulus (with respect to the Lebesgue measure) Mod Q ( A, B ) the capacity of ( A, B ) (see[Vuo88]).In X , the classical modulus are comparable to the combinatorial modulus in the follow-ing sense. Proposition 2.8 ([Haï09a, Prop B.2]) . Assume that X is equipped with an approximation { G k } k ≥ . For d > , let F be the set of curves in X of diameter larger than d . Then for k large enough one has Mod Q ( F , G k ) (cid:16) Mod Q ( F ) , if Mod Q ( F ) > and lim Mod Q ( F , G k ) = 0 if Mod Q ( F ) = 0 . n addition for any pair A, B of non-degenerate disjoint continua and for k large enoughone has Mod Q ( A, B, G k ) (cid:16) Mod Q ( A, B ) if Mod Q ( A, B ) > and lim Mod Q ( A, B, G k ) = 0 if Mod Q ( A, B ) = 0 . Note that this connection between combinatorial and classical modulus is only valid forthe dimension Q .Now we can define Loewner spaces. Definition 2.9.
We say that ( X, d, µ ) is a Q -Loewner space (or satisfies the Q -Loewnerproperty ) if there exists an increasing function φ : (0 , + ∞ ) −→ (0 , + ∞ ) such that for anypair of non-degenerate disjoint continua A and B in X one has: φ (∆( A, B ) − ) ≤ Mod Q ( F ( A, B )) . We also say that X satisfies the Loewner property or the classical Loewner property toavoid the confusion with the CLP.The control of the modulus from above is not required in this definition because it isautomatically provided by the structure of Q -AR space. Theorem 2.10 ([HK98, Lemma 3.14.]) . There exists a constant
C > such that thefollowing property holds. Let A and B be two non-degenerate disjoint continua. Let < r < R and x ∈ X be such that A ⊂ B ( x, r ) and B ⊂ X \ B ( x, R ) . Then Mod Q ( A, B ) ≤ C (cid:16) log Rr (cid:17) − Q . As a consequence there exists an increasing function ψ on (0 , + ∞ ) with lim t → ψ ( t ) = 0 ,such that for any pair of disjoint non-degenerate continua A and B Mod Q ( A, B ) ≤ ψ (∆ − ( A, B )) . More precisely, there exist some constants
K, C > such that ψ ( t ) = K (cid:16) log( t + C ) (cid:17) − Q for any t > .When X is a Loewner space, the asymptotic behavior of φ is described in [HK98,Theorem 3.6.]. For t small enough φ ( t ) ≈ log t , for t large enough φ ( t ) ≈ (log t ) − Q .As we will see in the sequel an essential difference between the combinatorial and theclassical modulus property is the importance of the dimension Q in the discussions aboutclassical modulus. 11 .4 First properties and examples In this section Z is a compact arcwise connected doubling metric space and X is a compact Q -AR metric space ( Q > ). First we recall a theorem and a conjecture that compare theCLP and the classical Loewner property. Theorem 2.11 ([BK13, Theorem 2.6.]) . If X is a compact Q -AR and Loewner metricspace, then X satisfies the combinatorial Q -Loewner property. The next conjecture is a main motivation for studying group boundaries that satisfythe CLP.
Conjecture 2.12 ([Kle06, Conjecture 7.5.]) . Assume that Z is quasi-Moebius homeomor-phic to the boundary of a hyperbolic group. If Z satisfies the CLP then it is quasi-Moebiushomeomorphic to a Loewner space. As announced we want to find and use the CLP on boundaries of hyperbolic groups.Quasi-isometries between hyperbolic spaces extend to quasi-Moebius homeomorphisms be-tween the boundaries, so it is fundamental to know how the Loewner property and theCLP behave under quasi-Moebius maps. These maps have been introduced by J. Vaïsäläin [Väi85]. We recall that in a metric space ( Z, d ) the cross-ratio of four distinct points a, b, c, d ∈ Z is [ a : b : c : d ] = d ( a, b ) d ( a, c ) · d ( c, d ) d ( b, d ) . For
Z, Z (cid:48) two metric spaces, an homeomorphism f : Z −→ Z (cid:48) is quasi-Moebius if thereexists an homeomorphism φ : [0 , + ∞ ) −→ [0 , + ∞ ) such that for any quadruple of distinctpoints a, b, c, d ∈ Z [ f ( a ) : f ( b ) : f ( c ) : f ( d )] ≤ φ ([ a : b : c : d ]) . If f is quasi-Moebius, as [ a : c : b : d ] = [ a : b : c : d ] − , f − : Z (cid:48) −→ Z is also quasi-Moebius. Theorem 2.13 ([BK13, Theorem 2.6.]) . If Z (cid:48) is quasi-Moebius homeomorphic to a compactspace Z satisfying the CLP, then Z (cid:48) also satisfies the CLP (with the same exponent). The Loewner property does not behave so well under quasi-Moebius maps. In particular,it is perturbed by a change of dimension whereas the CLP is not.
Theorem 2.14 ([Tys98]) . Let X and X (cid:48) be respectively Q -Loewner and Q (cid:48) -AR compactmetric spaces. Assume that X (cid:48) is quasi-Moebius homeomorphic to X . Then X (cid:48) is a Loewnerspace if and only if Q = Q (cid:48) . If we apply to X a snowflake transformation f (cid:15) : ( X, d ) −→ ( X, d (cid:15) ) , < (cid:15) < then dim H ( X, d (cid:15) ) = (cid:15) dim H ( Z, d ) . Such a transformation is a quasi-Moebius homeomorphismand Combining with the previous theorem we get the following fact.12 act 2.15. The Loewner property is not invariant under quasi-Moebius homeomorphisms.
However quasi-Moebius maps are the appropriate homeomorphisms to discuss betweenLoewner spaces.
Theorem 2.16 ([HK98]) . Let X and X (cid:48) be compact Q -regular Loewner spaces and let f : X −→ X (cid:48) be a homeomorphism. The following are equivalent1. f is quasi-Moebius,2. there exists C > such that C − · Mod Q ( F ) ≤ Mod Q ( f ( F )) ≤ C · Mod Q ( F ) for any set of curves F in X . The next proposition gives examples of spaces that do not satisfy the CLP.
Proposition 2.17 ([HK98] or [BK13, Theorem 2.6.]) . Assume that Z satisfies the CLPthen it has no local cut point, i.e no connected open subset is disconnected by removing apoint. Combining with the theorem of Bowditch (see [Bow98]) this proposition says that theboundary of a one-ended hyperbolic group that splits along a two-ended subgroup does notsatisfy the CLP.The first examples of spaces that satisfy the CLP are provided by Theorem 2.11 and byknown examples of Loewner spaces. The next examples are provided by [BK13].
Example 2.18.
1. The following spaces are Loewner spacesi) the Euclidean space R d for d ≥ , this result is due to C. Loewner for d ≥ (see[Loe59]),ii) any compact Riemannian manifold modeled by R d for d ≥ (see [HK98]),iii) visual boundaries of right-angled Fuchsian buildings (see [BP99]).2. The following spaces satisfy the CLP (see [BK13])i) the Sierpiński carpet and the n -dimensional Menger sponge embedded in the Eu-clidean space,ii) boundaries of Coxeter groups of various type: simplex groups, some prism groups,some highly symmetric groups and some groups with planar boundary. E satisfies the CLPFor Examples 2.18.2, we do not know if they are Loewner spaces. This provides a firstkind of interesting questions. Question 2.19.
Is any of the examples in 2.18.2. quasi-Moebius homeomorphic to aLoewner space?
Among these examples the Sierpiński carpet is the one that should be studied first asit should be the easiest one.Note that Example 2.18. .ii ) provides many examples of hyperbolic groups whoseboundaries are Loewner spaces. Indeed consider a group Γ that is acting geometrically(see Subsection 3.1) on the standard hyperbolic space H d for d ≥ . Then ∂ Γ is quasi-Moebius equivalent to S d − equipped with the standard spherical metric. Hence withExample 2.18 . .ii ) , ∂ Γ is a Loewner space.Now we consider the following general question: how the geometry of a hyperbolic spaceis determined by its boundary? The Cannon’s conjecture (see Conjecture 1.2) is a questionof this type. The notions of Loewner property and CLP have been fruitfully used by M.Bonk and B. Kleiner to approach this conjecture. By a theorem of D. Sullivan in [Sul82],Cannon’s conjecture is equivalent to the following in which the quasiconformal structure ofthe boundary is the main point. Conjecture 2.20 ([BK02, Conjecture 1.3.]) . If Γ is a hyperbolic group and ∂ Γ is homeo-morphic to S , then it is quasi-Moebius homeomorphic to the standard 2-sphere. If Conjecture 2.12 is true, the CLP would provide many interesting examples of Loewnerspaces. This motivates our second question.
Question 2.21.
Can we find new examples of compact metric spaces satisfying the CLP?
14n this article, we exhibit examples of boundaries of right-angled buildings of dimension3 and 4 that satisfy the CLP. These buildings have been studied by J. Dymara and D.Osajda who described the topology of the boundary.
Theorem 2.22 ([DO07]) . Let ∆ be a right-angled thick building whose associated Coxetergroup is a cocompact reflection group in H d . Then ∂ ∆ is homeomorphic to the universal ( d − -dimensional Menger space µ d − . Our interest in these buildings is inspired by Examples 2.18 . .iii ) and 2.18 . .ii ) . Wewill use some ideas from [BP00] and [BK13] where these examples are studied. Boundaries of hyperbolic groups are naturally endowed with metric structures that satisfy aproperty of self-similarity. This property implies Proposition 3.11 that will be used severaltimes in proving the main theorem. Intuitively, we can say that this proposition is a toolto enlarge sets of small curves while controlling the modulus.In this section, we explain how the boundary of a hyperbolic group can be seen asapproximately self-similar spaces. Then, we recall the connection between the combinato-rial modulus and the conformal dimension of the boundary. Finally, we give a sufficientcondition for the boundary to satisfy the CLP.Most of this section is a review of [BK13, Section 3 and 4] to which we refer for details.
For details concerning hyperbolic groups and spaces, we refer to [CDP90], [GdlH90] or[KB02]. Let ( X, d ) be a proper geodesic metric space. We say that a finitely generatedgroup Γ acts geometrically on X , if: • Γ < Isom ( X ) , • Γ acts cocompactly, • Γ acts properly discontinuously.We say that X is hyperbolic (in the sense of Gromov) if there exists a constant δ > such that for every geodesic triangle [ x, y ] ∪ [ y, z ] ∪ [ z, x ] ⊂ X and every p ∈ [ x, y ] , one has dist( p, [ y, z ] ∪ [ z, x ]) ≤ δ. A finitely generated group that acts geometrically on a hyperbolic space X is called a hyperbolic group . In this case, the Cayley graph of a hyperbolic group is a hyperbolic space.15rom now on, let X be a hyperbolic space with a fixed base point x and let Γ be ahyperbolic group acting geometrically on X . Let ∂X be the set of equivalence classes ofgeodesic rays where two geodesic rays γ, γ (cid:48) : [0 , + ∞ ) −→ X are equivalent if and only if:there exists K > such that d ( γ ( t ) , γ (cid:48) ( t )) ≤ K for any t ∈ [0 , + ∞ ) .Thanks to the hyperbolicity condition, we can restrict to the set of geodesic rays startingfrom x . We can equip ∂X and X ∪ ∂X with topologies which make them compact spaces.In this setting X , is dense in X ∪ ∂X and ∂X is called the boundary of X . Moreover wecan equip ∂X with a family of visual metric . A metric δ ( · , · ) is visual if there exist twoconstants A ≥ and α > such that for all ξ, ξ (cid:48) ∈ ∂X : A − e − α(cid:96) ≤ δ ( ξ, ξ (cid:48) ) ≤ A e − α(cid:96) , where (cid:96) is the distance from x to a geodesic line ( ξ, ξ (cid:48) ) . In such a situation we also write δ ( ξ, ξ (cid:48) ) (cid:16) e − α(cid:96) . The action of Γ on X extends naturally to ( ∂X, δ ) and elements of Γ are uniform quasi-Moebius homeomorphisms of the boundary. Moreover, if ∂ Γ is also equipped with a visualmetric, the homeomorphism ∂ Γ −→ ∂X induced by the orbit map g ∈ Γ −→ gx ∈ X isquasi-Moebius.The following definition is a generalization of the classical notion of self-similar space. Definition 3.1.
A compact metric space ( Z, d ) is called approximately self-similar if thereexists a constant L ≥ such that for every r -ball B with < r < diam Z , there exists anopen subset U ⊂ Z which is L -bi-Lipschitz homeomorphic to the rescaled ball ( B, r d ) . The definition and proposition that follow say that the boundary of a hyperbolic groupis an approximately self-similar space and that this structure is linked to the action of thegroup on its boundary.
Definition 3.2.
Let Γ be a hyperbolic group. A metric d on ∂ Γ is called a self-similarmetric if there exists a hyperbolic space X on which Γ acts geometrically, such that d isthe preimage of a visual metric on ∂X by the canonical quasi-Moebius homeomorphism ∂ Γ −→ ∂X . Proposition 3.3 ([BK13, Proposition 3.3.]) . The space ∂ Γ equipped with a self-similarmetric is doubling and is an approximately self-similar space, the partial bi-Lipschitz mapsbeing restrictions of group elements. Moreover, Γ acts on ( ∂ Γ , d ) by (non-uniformly) bi-Lipschitz homeomorphisms. .2 Combinatorial modulus and conformal dimension In this subsection we present the connection between combinatorial modulus and the con-formal dimension in approximately self-similar spaces.Let Z be an arcwise connected approximately self-similar metric space. For us Z willbe the boundary of a hyperbolic group. Let { G k } k ≥ be a κ -approximation of Z and d bea small constant compared with diam Z and with the constant of self-similarity.Let F denote the set of all the curves in Z of diameter larger than d . In [BK13], it isproved that the properties of the combinatorial modulus are contained in the asymptoticbehavior of Mod p ( F , G k ) . This point is explained in Subsection 3.3. Here we write M k :=Mod p ( F , G k ) .The following proposition allows to define a critical exponent that is related to theconformal dimension of Z . Proposition 3.4.
There exists p ≥ such that for p ≥ p the modulus M k goes to zeroas k goes to infinity.Proof. Let { G k } k ≥ be a κ -approximation of Z . According to the doubling condition andthe definition of an approximation, there exists an integer N (cid:48) such that each element v ∈ G k is covered by at most N (cid:48) elements of G k +1 . As a consequence, if K > is the cardinality G , then G k ≤ K · N (cid:48) k for any k ≥ . Moreover, as we saw in the proof of Proposition 2.4, there exists a constant K (cid:48) > suchthat the constant function ρ : v ∈ G k −→ ρ ( v ) = K (cid:48) · − k ∈ [0 , + ∞ ) is F -admissible.As a consequence M k ≤ C · ( N (cid:48) p ) k , where C is a positive constant. Thus, for p large enough, M k goes to zero.According to this proposition the next definition makes sense. Definition 3.5.
The critical exponent Q associated with the curve family F is defined asfollows Q = inf { p ∈ [1 , + ∞ ) : lim k → + ∞ Mod p ( F , G k ) = 0 } . We notice that for k ≥ fixed the function : p (cid:55)−→ Mod p ( F , G k ) is non-increasing.Hence Mod p ( F , G k ) goes to zero for p > Q .This critical exponent is related to the conformal dimension, which provides anothermotivation to study combinatorial modulus. The conformal dimension has been introducedby P. Pansu in [Pan89]. It is an important property of the conformal structure of theboundary of a hyperbolic group. In particular, it is invariant under quasi-Moebius maps.17n the following, H d ( · ) and dim H ( Z, d ) respectively denote the Hausdorff measure andthe Hausdorff dimension of Z equipped with d . The Ahlfors-regular conformal gauge of ( Z, d ) is defined as follows J c ( Z, d ) := { ( Z (cid:48) , δ ) : ( Z (cid:48) , δ ) is AR and quasi-moebius homeomorphic to ( Z, d ) } . Definition 3.6.
Let ( Z, d ) be a compact metric space. The Ahlfors-regular conformaldimension of ( Z, d ) is Confdim(
Z, d ) := inf { dim H ( Z (cid:48) , δ ) : ( Z (cid:48) , δ ) ∈ J c ( Z, d ) } . In the rest of the paper we will simply call it the conformal dimension .In [KK], S. Keith and B. Kleiner proved that the combinatorial modulus are related tothe conformal dimension. The proof of the following theorem may be found in [Car11].
Theorem 3.7 ([KK] or [Car11, Theorem 4.5.]) . The critical exponent Q (see Definition3.5) is equal to Confdim(
Z, d ) . The definition of the conformal dimension, Combining with basic topology give thefollowing inequalities: dim T ( Z ) ≤ Confdim(
Z, d ) ≤ dim H ( Z, d ) , where dim T ( Z ) is the topological dimension of Z .The following theorem due to J. Tyson makes a connection between the conformaldimension and the Loewner property. Theorem 3.8 ([MT10, Corollary 4.2.2.]) . Let
Q > and X be a Q -regular and Q -Loewnerspace, then Confdim( X ) = Q . Example 3.9.
It has been proved independently by B. Kleiner and in [KL04] that theEuclidean metric on the Sierpiński carpet does not realize the conformal dimension. Asa consequence the Sierpiński carpet equipped with this metric does not satisfy the Loewnerproperty. However it satisfies the CLP (see Example 2.18).
Again, Cannon’s conjecture has been an important motivation for studying the confor-mal dimension of the boundary of a hyperbolic group. In particular in [BK05] it is provedthat Conjecture 1.2 is equivalent to the following.
Conjecture 3.10. If Γ is a hyperbolic group and ∂ Γ is homeomorphic to S , then Confdim( ∂ Γ) is attained by a metric in J c ( ∂ Γ) . .3 How to prove the CLP Now we give the sufficient condition that will be used in this article to exhibit some examplesof groups with a boundary that satisfies the CLP.Let Z be an arcwise connected approximately self-similar metric space and let { G k } k ≥ be a κ -approximation of Z .The following proposition says that the combinatorial modulus are preserved by thebi-Lipschitz homeomorphism coming from the approximately self-similar structure. Thisproposition will be used several times in Sections 6 to 10 to compare the modulus of a setof small curves with the modulus of a set of curves of diameter larger than a fixed constant. Proposition 3.11.
Let B be a ball in ∂ Γ such that diam B < . Let g ∈ Γ be the local L -bi-Lipschitz homeomorphism that rescales B (given by Definition 3.3). Let F be a set ofcurves contained in λB for λ < . Then there exist (cid:96) ∈ N , and D > depending only on L and on the doubling constant of ∂ Γ such that the following property holds.If k ≥ is large enough so that { v ∈ G k : γ ∩ v (cid:54) = ∅ for some γ ∈ F } ⊂ { v ∈ G k : v ⊂ B } , then D − · Mod p ( g F , G k ) ≤ Mod p ( F , G k + (cid:96) ) ≤ D · Mod p ( g F , G k ) , where g F = { gγ : γ ∈ F } .Proof. Let k ≥ be large enough so that, if γ ∩ v (cid:54) = ∅ with γ ∈ F and v ∈ G k , then v ⊂ B .Let d = diam B and let (cid:96) ∈ N denote the integer satisfying − ( (cid:96) +1) < d ≤ − (cid:96) . Let v ∈ G k + (cid:96) such that v ⊂ B and assume B ( ξ, κ − − ( k + (cid:96) ) ) ⊂ v ⊂ B ( ξ, κ − ( k + (cid:96) ) ) . Then B ( gξ, ( Lκ ) − − k ) ⊂ gv ⊂ B ( gξ, Lκ − k ) . We write G (cid:48) k ∩ gB = { gv : v ∈ G k + (cid:96) , v ⊂ B } . Then G (cid:48) k ∩ gB is a κ (cid:48) -approximation of gB on scale k , with κ (cid:48) depending only on κ and L . As the curves of F are strictly containedin B we obtain the following equality Mod p ( F , G k + (cid:96) ) = Mod p ( g F , G (cid:48) k ∩ gB ) . Thanks to Proposition 2.5, there exists
D > such that D − · Mod p ( g F , G k ) ≤ Mod p ( g F , G (cid:48) k ∩ gB ) ≤ D · Mod p ( g F , G k ) , and the proposition follows. 19ow we fix d > a small constant compared with diam Z and with the constant ofself-similarity. More precisely it must be small enough so that any non-constant curve in Z may be rescaled to a curve of diameter larger than d by self-similarity. For us, Z willbe the boundary of a hyperbolic group and d will depend on the hyperbolicity constant.In the following, F is the set of all the curves in Z of diameter larger than d .Again, we use the letter Q to designate the critical exponent of Definition 3.5. We recallthat if η is a curve in Z , then U (cid:15) ( η ) denotes the (cid:15) -neighborhood of η for the C -topology.This means that a curve η (cid:48) ∈ U (cid:15) ( η ) if and only if there exists s : t ∈ [0 , −→ [0 , aparametrization of η such that for any t ∈ [0 , one has d ( η ( s ( t )) , η (cid:48) ( t )) < (cid:15) .The following proposition gives the sufficient conditions for Z to satisfy the CLP thatwill be used in proving the main theorem. Proposition 3.12 ([BK13, Proposition 4.5.]) . Let Z be an arcwise connected approximatelyself-similar metric space. Let { G k } k ≥ be a κ -approximation of Z and d be a small constantcompared with diam Z and with the constant of self-similarity. Let F denote the set of allthe curves in Z of diameter larger than d .For p = 1 , we assume that Mod p ( F , G k ) is unbounded. For p ≥ , we assume that forevery non-constant curve η ⊂ Z and every (cid:15) > , there exists C = C ( p, η, (cid:15) ) such that forevery k ∈ N : Mod p ( F , G k ) ≤ C · Mod p ( U (cid:15) ( η ) , G k ) . Suppose furthermore when p belongs to a compact subset of [1 , + ∞ ) the constant C may bechosen independent of p . Then Z satisfies the combinatorial Q -Loewner property. Before going into details about boundaries of right-angled buildings, we give a sketch ofproof of the main theorem. In this section, D is the right-angled dodecahedron in H orthe right-angled 120-cell in H . We write W D for the hyperbolic reflection group generatedby the faces of D . The main theorem of this article may be stated as follows. Theorem 4.1 (Corollary 10.3) . Let Γ be the graph product of constant thickness q ≥ andof Coxeter group W D . Then ∂ Γ equipped with a visual metric satisfies the CLP. As announced, we will verify that ∂ Γ satisfies the hypothesis of Proposition 3.12. Toprove that Mod ( F , G k ) is unbounded, it is enough to see that for every N ∈ N there exist N disjoint curves in ∂ Γ . Indeed, this implies that for k ≥ large enough Mod ( F , G k ) > N . To follow curves to control the modulus.
For p > , we want to prove that thecurves of ∂ Γ satisfy the following property. ( P ) : For (cid:15) > , there exists a finite set F of bi-Lipschitz maps f : ∂ Γ −→ ∂ Γ such that for any curve γ ∈ F and any curve η in ∂ Γ , thesubset (cid:83) f ∈ F f ( γ ) of ∂ Γ contains a curve that belongs to U (cid:15) ( η ) .20here U (cid:15) ( η ) denotes the (cid:15) -neighborhood of η for the C distance (see Subsection 1.4 fordetails). Intuitively, ( P ) holds if from any curve γ we can follow any other curve η usingbi-Lipschitz maps. The following computation shows that property ( P ) implies the desiredinequality. Proposition 4.2. If Mod ( F , G k ) is unbounded, then property ( P ) implies the CLP.Proof. Let η be a curve in ∂ Γ and (cid:15) > . Fix ρ a U (cid:15) ( η ) -admissible function. The inequalityrequired by the hypothesis of Proposition 3.12 is obtained if we find a constant K > independent of the scale k and a F -admissible function ρ (cid:48) such that M p ( ρ (cid:48) ) ≤ K · M p ( ρ ) .Let F be the set of bi-Lipschitz maps given by the property ( P ) . We assume, withoutloss of generality that F contains F − . We define the function ρ (cid:48) : G k −→ [0 , + ∞ ) by: ( ∗ ) ρ (cid:48) ( v ) = (cid:88) f ∈ F (cid:88) fw ∩ v (cid:54) = ∅ ρ ( w ) . Let γ ∈ F and θ be a curve contained in (cid:83) f ∈ F f γ ⊂ ∂ Γ such that θ ∈ U (cid:15) ( η ) . Then L ρ (cid:48) ( γ ) = (cid:88) f ∈ F (cid:88) v ∩ γ (cid:54) = ∅ (cid:88) fw ∩ v (cid:54) = ∅ ρ ( w ) ≥ (cid:88) f ∈ F (cid:88) w ∩ fγ (cid:54) = ∅ ρ ( w ) . On the other hand L ρ ( θ ) ≤ (cid:88) f ∈ F L ρ ( f γ ) = (cid:88) f ∈ F (cid:88) w ∩ fγ (cid:54) = ∅ ρ ( w ) . Hence L ρ (cid:48) ( γ ) ≥ L ρ ( θ ) and ρ (cid:48) is F -admissible.Then the number of terms in the right-hand side of the definition ( ∗ ) is bounded by aconstant N depending only on F , the bi-Lipschitz constants of the elements of F , andthe doubling constant of ∂ Γ . Therefore by convexity M p ( ρ (cid:48) ) = (cid:88) v ∈ G k (cid:16) (cid:88) f ∈ F (cid:88) fw ∩ v (cid:54) = ∅ ρ ( w ) (cid:17) p ≤ N p − · (cid:88) v ∈ G k (cid:88) f ∈ F (cid:88) w ∩ fv (cid:54) = ∅ ρ ( w ) p ≤ N p · F · M p ( ρ ) . Note that this idea of following curves may be used to obtain an inequality between themodulus of any pair of sets of curves.
The issue of parabolic limit sets.
Since Γ acts on ∂ Γ by bi-Lipschitz homeomorphisms,it is natural to try to follow curves by using the elements of Γ . However, in right-angledbuildings some curves may be contained in parabolic limit sets (boundaries of residues). Aswe will see in Example 6.8, these curves are obstacles to proving the property ( P ) by usingthe elements of Γ .To solve this problem we start by showing that Mod p ( F , G k ) is determined by thecombinatorial modulus of the sets of all the curves contained in a parabolic limit set. Thisis done at the beginning of the proof of Theorem 8.13.21 ollowing curves inside parabolic limit sets. Then inside the parabolic limit set ∂P it is possible to follow curves. An analogue of property ( P ) inside the parabolic limit sets isproved in Proposition 6.11. From this property we can obtain Theorem 6.12. This theoremis the first major step toward the proof. Essentially it says that the combinatorial modulusof F δ,r ( ∂P ) is controlled by any curve contained in ∂P . Controlling the modulus in ∂ Γ by the modulus in the boundary of an apartment. The second major step in the proof is to use the building structure to reduce the problemin ∂ Γ to a problem in the boundary of an apartment i.e in ∂W D . This is done by Theorem8.9. Essentially, this theorem says that the modulus of a curve family in ∂ Γ is controlledby a weighted modulus defined in the boundary of an apartment. The idea used to provethis is that, from the point of view of the modulus, ∂ Γ can be seen as the direct product ofthe boundary of an apartment by a finite set whose cardinality depends on the scale. Conclusion of the proof thanks to the symmetries of D . Now, thanks to Theorems6.12 and 8.9, we arrive at a point where the modulus of F δ,r ( ∂P ) , and thus of F , iscontrolled by some modulus of the parabolic limit sets of W D . The idea we use to concludeis that the symmetries of D extends to the boundary of W D . Combining with the elementsof the groups, these symmetries permit us to follow curves in ∂W D . As a consequencewe obtain a strong control of the modulus of the parabolic limit sets in ∂W D and we cancomplete the proof. The aim of this article is to study combinatorial modulus on boundaries of hyperbolicbuildings. Below we set up the context about hyperbolic buildings that will be used untilthe end of this article. In particular, we discuss the geometry of locally finite right-angledhyperbolic buildings.For details concerning the notions recalled in this section, we refer to [Tit74], [Ron89],or [AB08]. Concerning the Davis realization, we can refer to [Dav08, Chapter 8] or to[Mei96] for an example of the Davis construction along with suggestive pictures. Bellow S = { s , . . . , s n } is a fixed finite set. Following the definition of J. Tits, a chamber system X over S is a set endowed with afamily of partitions indexed by S . The elements of X are called chambers .Hereafter X is a chamber system over S . For s ∈ S , two chambers c, c (cid:48) ∈ X are saidto be s -adjacent if they belong to the same subset of X in the partition associated with s . Then we write c ∼ s c (cid:48) . Usually, omitting the type of adjacency we refer to adjacent chambers and we write c ∼ c (cid:48) . Note that any chamber is adjacent to itself.22 morphism f : X −→ X (cid:48) between two chamber systems X, X (cid:48) over S is a map thatpreserves the adjacency relations. A bijection of X that preserves the adjacency relations iscalled an automorphism and we designate by Aut( X ) the group of automorphisms of X . A subsystem of chamber Y of X is a subset Y ⊂ X such that the inclusion map is a morphismof chamber systems.We call gallery , a finite sequence { c k } k =1 ,...,(cid:96) of chambers such that c k ∼ c k +1 for k = 1 , . . . , (cid:96) − . The galleries induce a metric on X . Definition 5.1.
The distance between two chambers x and y is the length of the shortestgallery connecting x to y . We use the notation d c ( · , · ) for this metric over X . A shortest gallery between twochambers is called minimal .Let I ⊂ S . A subset C of X is said to be I -connected if for any pair of chambers c, c (cid:48) ∈ C there exists a gallery c = c ∼ · · · ∼ c (cid:96) = c (cid:48) such that for any k = 1 , . . . , (cid:96) − , thechambers c k and c k +1 are i k -adjacent for some i k ∈ I . Definition 5.2.
The I -connected components are called the I -residues or the residues oftype I . The cardinality of I is called the rank of the residues of type I . The residues ofrank are called panels . The following notion of convexity is used in chamber systems.
Definition 5.3.
A subset C of X is called convex if every minimal gallery whose extremitiesbelong to C is entirely contained in C . The convexity is stable by intersection and for A ⊂ X , the convex hull of A is thesmallest convex subset containing A . In particular, convex subsets of X are subsystems.The following example is crucial because it will be used to equip Coxeter groups and graphproducts with structures of chamber systems (see Definition 5.7 and Theorem 5.16). Example 5.4.
Let G be a group, B a subgroup and { H i } i ∈ I a family of subgroups of G containing B . The set of left cosets of H i /B defines a partition of G/B . We denote by C ( G, B, { H i } i ∈ I ) this chamber system over I . This chamber system comes with a naturalaction of G . The group G acts by automorphisms and transitively on the set of chambers. A Coxeter matrix over S is a symmetric matrix M = { m r,s } r,s ∈ S whose entries are elementsof N ∪ {∞} such that m s,s = 1 for any s ∈ S and { m r,s } ≥ for any r, s ∈ S distinct. Let M be a Coxeter matrix. The Coxeter group of type M is the group given by the followingpresentation W = (cid:104) s ∈ S | ( rs ) m r,s = 1 for any r, s ∈ S (cid:105) . We call special subgroup a subgroup of W of the form W I = (cid:104) s ∈ I | ( rs ) m r,s = 1 for any r, s ∈ I (cid:105) with I ⊂ S. efinition 5.5. We call parabolic subgroup a subgroup of W of the form wW I w − where w ∈ W and I ⊂ S . An involution of the form wsw − for w ∈ W and s ∈ S is called a reflection . Example 5.6.
Let X d = S d , E d or H d . A Coxeter polytope is a convex polytope of X d suchthat any dihedral angle is of the form πk with k not necessarily constant. Let D be a Coxeterpolytope and let σ , . . . , σ n be the codimension 1 faces of D . We set M = { m i,j } i,j =1 ,...,n the matrix defined by m i,i = 1 , m i,j = ∞ if σ i and σ j do not meet in a codimension 2 face,and m i,j = k if σ i and σ j meet in a codimension 2 face and πk is the dihedral angle between σ i and σ j .Then a theorem of H. Poincaré (see [GP01, Theorem 1.2.]) says that the reflectiongroup of X d generated by the codimension 1 faces of D is a discrete subgroup of Isom( X d ) and is isomorphic to the Coxeter group of type M . Definition 5.7.
With the notation introduced in Example 5.4, the
Coxeter system asso-ciated with W is the chamber system over S given by C ( W, { e } , { W { s } } s ∈ S ) . We use thenotation ( W, S ) to denote this chamber system. The chambers of ( W, S ) are the elements of W and two distinct chambers w, w (cid:48) ∈ W are s -adjacent if and only if w = w (cid:48) s . For I ⊂ S , notice that for any I -residue R in ( W, S ) there exists w ∈ W such that, as a set R = wW I . Again W is a group of automorphismsof ( W, S ) that acts transitively on the set of chambers.Hereafter ( W, S ) is a fixed Coxeter system. Example 5.8.
In the case of Example 5.6, the chamber system associated with W is realizedgeometrically by the tilling of X d by copies of the polytope D . Two chambers are adjacentin ( W, S ) if and only if the corresponding copies of D in X d share a codimension 1 face. ( W, S ) To a Coxeter group W , M.W. Davis associates a cellular complex D called the Davischamber . In the particular case of reflection groups (see Examples 5.6 and 5.8) the DavisChamber is the Coxeter polytope.We recall that S = { s , . . . , s n } is a set of generators of W . Let S (cid:54) = S be the collectionof proper subsets of S . We denote by S f the set of proper subsets F (cid:32) S such that W F is finite.By [Dav08, Appendix A], a poset admits a geometric realization which is a simplicialcomplex. This complex is such that the inclusion relations between cells represent thepartial order. We denote by D the Davis chamber which is the geometric realization of theposet S f . In the following we give details of this construction.Let ∆ n − be the standard ( n − -simplex and label the codimension 1 faces of ∆ n − with distinct elements of S . Then a codimension k face σ of ∆ n − is associated with a24 ype i.e a subset I ⊂ S of cardinality k . In this case, we write σ I for the face of type I . Equivalently, we can say that each vertex of the barycentric subdivision of ∆ n − isassociated with a subset of S . Combining with the fact that the empty set is associatedwith the barycenter of the whole simplex, we get a bijection between the vertices of thebarycentric subdivision and S (cid:54) = S . Hence a vertex in the barycentric subdivision is designatedby ( s i ) i ∈ K for K ⊂ { , . . . , n } . Using this identification, let T be the subgraph of the -skeleton of the barycentric subdivision of ∆ n − defined as follows: • T (0) = S (cid:54) = S , • the vertices ( s i ) i ∈ I and ( s j ) j ∈ J , with J ≥ I , are adjacent if and only if I ⊂ J and I = J − .In the following definition, for k ≥ we call a k -cube, a CW-complex that is isomorphic,as a cellular complex, to the Euclidean k -cube [0 , k . In particular, it is not necessary toequip these cubes with a metric for the purpose of this chapter. Definition 5.9.
The -skeleton D (1) of the Davis chamber is the full subgraph of T gener-ated by the elements of S f . The Davis chamber is obtained from D (1) by attaching a k -cubeto every subgraph that is isomorphic to the -skeleton of a k -cube. By construction, D ⊂ ∆ n − . We call maximal faces of D the subsets of the form σ ∩ D where σ is a codimension 1 face of ∆ n − . Likewise, for I ⊂ S , the face of D of type I is D ∩ σ I . Example 5.10.
In the case of Example 5.6, the Davis chamber is combinatorially identi-fied with the Coxeter polytope. So if we equip D with the appropriate metric (Euclidean,spherical, or hyperbolic) we recover the Coxeter polytope. Buildings are singular spaces defined by J. Tits. We may view them as higher dimensionalanalogues of trees. Hereafter ( W, S ) is a fixed Coxeter system. Definition 5.11 ([Tit74, Definition 3.1.]) . A chamber system ∆ over S is a building oftype ( W, S ) if it admits a maximal family A p (∆) of subsystems isomorphic to ( W, S ) , calledapartments, such that • any two chambers lie in a common apartment, • for any pair of apartments A and B , there exists an isomorphism from A to B fixing A ∩ B . An immediate consequence of this definition is the existence of retraction maps of thebuilding onto the apartments. 25 efinition 5.12.
Let x ∈ ∆ and A ∈ A p (∆) . Assume that x is contained in A . We call retraction onto A centered at x the map π A,x : ∆ −→ A defined by the following property.For c ∈ ∆ , there exists a chamber π A,x ( c ) ∈ A such that for any apart-ment A (cid:48) containing x and c , for any isomorphism f : A (cid:48) −→ A that fixes A ∩ A (cid:48) , we have f ( c ) = π A,x ( c ) Hereafter, ∆ is a fixed building of type ( W, S ) . The building ∆ is called a thin (resp. thick ) building if any panel contains exactly two (resp. at least three) chambers. Notethat thin buildings are Coxeter systems. Let G denote a finite simplicial graph i.e G (0) is finite, each edge has two different vertices,and G contains no double edge. We denote by G (0) = { v , . . . , v n } the vertices of G . Iffor i (cid:54) = j , the corresponding vertices v i , v j are connected by an edge, we write v i ∼ v j . Afinite cyclic group G i = (cid:104) s i (cid:105) of order q i ≥ is associated with each v i ∈ G (0) and we set S = { s , . . . , s n } . Throughout this article, we assume that n ≥ and that G has at leastone edge. Definition 5.13.
The graph product given by ( G , { G i } i =1 ,...,n ) is the group defined by thefollowing presentation Γ = (cid:104) s i ∈ S | s q i i = 1 , s i s j = s j s i if v i ∼ v j (cid:105) . Example 5.14.
If the graph G is fixed with q i ∈ { , , . . . , } , graph products are groupsbetween right-angled Coxeter groups (see [Dav08]) and right-angled Artin groups (see[Cha07]). If the integers { q i } i =1 ,...,n are fixed and we add edges to the graph starting froma graph with no edge, those groups are groups between free products and direct products ofcyclic groups. From now on, we fix a graph product Γ associated with the pair ( G , { G i } i =1 ,...,n ) . Byanalogy with Definition 5.5, we define parabolic subgroups in Γ . Definition 5.15.
The subgroup of Γ generated by a subset I ⊂ S is denoted by Γ I and asubgroup of the form g Γ I g − , with g ∈ Γ , is called a parabolic subgroup . Let W be the graph product defined by the pair ( G , { Z / Z } i =1 ,...,n ) . This graph productis isomorphic to the right-angled Coxeter group of type M = { m i,j } i,j =1 ,...,n defined by : m i,j = 2 if v i ∼ v j and m i,j = ∞ if v i (cid:28) v j .Throughout this article, W denotes this Coxeter group canonically associated with Γ and ( W, S ) is the Coxeter system associated with W . Theorem 5.16 ([Dav98, Theorem 5.1.]) . Let ∆ be the chamber system C (Γ , { e } , { Γ { s } } s ∈ S ) (see Example 5.4). Then ∆ is a building of type ( W, S ) . ∆ denotes the right-angled building associated with Γ by the preceding the-orem. In Subsection 5.6, we describe the Davis complex associated with this building.We notice that Γ is infinite if and only if the graph G contains two distinct vertices v i , v j such that v i (cid:28) v j . A criterion of M. Gromov allows J. Meier to prove that an infinite graphproduct Γ is hyperbolic if and only if in G any circuit of length contains a chord (see[Mei96]). For an infinite hyperbolic graph product, a necessary and sufficient condition isgiven in [DM02] for ∂ Γ to be arcwise connected (see Theorem 6.16 bellow). This conditioninvolves only the graph G . In the rest of this paper, we will assume that Γ is infinitehyperbolic with arcwise connected boundary ∂ Γ . Γ To a graph product Γ , M.W. Davis associates a cellular complex Σ called the Davis complex .This complex is a metric space on which Γ acts geometrically. In the particular case ofreflection groups (see Examples 5.6 and 5.8) the Davis Complex is X d tilled by the Coxeterpolytopes. This complex is also called the geometric realization of the building ∆ . Belowwe introduce the Davis complex associated with Γ . Again we refer to [Mei96] for an examplealong with suggestive pictures.Let D be the Davis chamber associated with W as in Subsection 5.3. Again a face of D is associated with a type I ⊂ S . For x ∈ D , if I is the type of the face containing x inits interior, we set Γ x := Γ I . To the interior points of D we associate the trivial group Γ ∅ .Now we can define the Davis complex : Σ = D × Γ / ∼ with ( x, g ) ∼ ( y, g (cid:48) ) if and only if x = y and g − g (cid:48) ∈ Γ x . We study the building ∆ through it geometric realization Σ and we briefly recall whatthis means.A chamber of Σ is a subset of the form [ D × { g } ] with g ∈ Γ . Two chambers areadjacent if and only if they share a maximal face. For a subset E ⊂ Σ we designate by Ch( E ) the set of chambers contained in E . Equipped with this chamber system structure, Σ is isomorphic to ∆ . In particular, the set of apartments in Σ is designated by A p (Σ) .Then the left action of Γ on itself induces an action on Σ . For γ ∈ Γ and [( x, g )] ∈ Σ weset γ [( x, g )] := [( x, γg )] . Moreover this action induces a simply transitive action of Γ on Ch(Σ) . Naturally this action is also isometric for d c ( · , · ) . Example 5.17.
In the case of Example 5.6, if we equip D with the appropriate metric wesee that the Davis complex is realized by the tilling of X d by D . We call base chamber of Σ , denoted by x , the chamber [ D × { e } ] . For g ∈ Γ , as [ D × { g } ] is the image of x under g , we designate this chamber by gx . Below we present some basic27ools used to describe the structure of Σ . In particular, we extend to Σ some definitionsand properties that have been used in Coxeter systems.The notion of walls in a Coxeter system extends to right-angled buildings. Definition 5.18.
1. We call building-wall in Σ the subcomplex M stabilized by a non-trivial isometry r = gs α g − with g ∈ Γ , s ∈ S , α ∈ Z and s α (cid:54) = e . The isometry r is called a rotationaround M . We denote by M (Σ) the set of all the building-walls of Σ .2. Let M be a building-wall associated with a rotation r ∈ Γ . For x ∈ Ch(Σ) we say that M is along x if r ( x ) is adjacent to x . With the notation of the definition , the term s ∈ S is called the type of the building-wall M and of the rotation r . We remark that in the graph product Γ two elements of S thatare conjugate are equal so the type is uniquely defined. Indeed if s = gs (cid:48) g − with g ∈ Γ , s, s (cid:48) ∈ S , let M be the building-wall associated with the rotation s . Then we observe thatfrom the base chamber x we can reach the chamber gx by successive rotations aboutfaces that are all orthogonal to M . In other word g is a product of elements of S that allcommute with s . Thus s = s (cid:48) . Clearly with the notation of the definition, the building-wall M is fixed by any rotation gs α (cid:48) g − with s α (cid:48) (cid:54) = e .We say that the building-wall M is non-trivial if it contains more than one point.A non-trivial building-wall M may be equipped with a building structure. Indeed, if s i is the type of M , associated with v i ∈ G (0) , we write I = { j : v j ∼ v i , v j (cid:54) = v i } and V = { v j ∈ G (0) : j ∈ I } . Then if G V is the full subgraph generated by V , we can check that, M is isomorphic to the geometric realization of the graph product ( G V , { Z /q i Z } i ∈ I ) . TheDavis chamber of this geometric realization is the maximal face of type s i of D . Moreoverbuilding-walls also divide Σ in isomorphic connected components. In the case of infinitedimension 2 buildings, the building-walls are trees and thus they have been called trees-walls by M. Bourdon and H. Pajot in [BP00]. These explain our terminology. Definition 5.19.
Let M be a building-wall of type s and let r ∈ Γ be a rotation around M .A dial of building bounded by M is the closure in Σ of a connected component of Σ \ M .We denote by D (Σ) the set of all the dials of building of Σ . This definition implies the following fact.
Fact 5.20.
Let M be a building-wall of type s . Assume that s is of finite order q . Then Σ \ M consists of q connected components. We designate by D ( M ) , D ( M ) , . . . , D q − ( M ) these dials of building, with the convention that x ⊂ D ( M ) . In this setting, for any i = 0 , . . . , q − , if y ∈ Ch( D i ( M )) then Ch( D i ( M )) = { x ∈ Ch(Σ) : d c ( y, x ) < d c ( y, rx ) } . inally r permutes D ( M ) , D ( M ) , . . . , D q − ( M ) .For a building-wall associated with a type s ∈ S of infinite order, the analogous propertyholds. In thin right-angled buildings, in particular in apartments, building-walls are called walls and dials of building are called half-spaces . For A ∈ A p (∆) we write M ( A ) for theset of all the walls and H ( A ) for the set of all the half-spaces of A .The building-walls in Σ and the walls in the apartments are closely related. Fact 5.21. i) For A ∈ A p (∆) and x ∈ Ch( A ) we have M ( A ) = { M ∩ A : M ∈ M (Σ) and M ∩ A (cid:54) = ∅} , = { π A,x ( M ) : M ∈ M (Σ) and M ∩ A (cid:54) = ∅} , = { π A,x ( M ) : M ∈ M (Σ) } . ii) For A ∈ A p (∆) and x ∈ Ch( A ) we have M (Σ) = (cid:91) m ∈M ( A ) C ( π − A,x ( m )) , where C ( π − A,x ( m )) denotes the set of all the connected components of π − A,x ( m ) . Building-walls (resp. walls) bound dials of buildings (resp. half-spaces) so a similar factholds for dials of building and half-spaces.We use the following terminology to describe a building-wall relatively to some cham-bers.
Definition 5.22.
Let M ∈ M (Σ) and E, F ⊂ Σ .i) We say that M crosses E if E \ M has at least two connected components.ii) We say that the building-wall M separates E and F if the interior of E and F areentirely contained in two distinct connected components of Σ \ M . The metric over the chambers is determined by the building-wall structure.
Proposition 5.23 ([Cla13, Proposition 5.8] ) . Let x and x be two chambers. If we denote d c ( · , · ) the metric on the chamber system, then d c ( x , x ) = { M ∈ M (Σ) : M seperates x and x } . In a right-angled building it appears that two distinct building-walls are either orthog-onal or do not intersect. This explains the following terminology and notation.
Notation.
Let M and M (cid:48) be two distinct building-walls. ) if M ∩ M (cid:48) (cid:54) = ∅ we write M ⊥ M (cid:48) and we say that M is orthogonal to M (cid:48) ,ii) if M ∩ M (cid:48) = ∅ we write M (cid:107) M (cid:48) and we say that M is parallel to M (cid:48) . Clearly, if
D, D (cid:48) ∈ D (Σ) are bounded by
M, M (cid:48) ∈ M (Σ) with if M ⊥ M (cid:48) , then D ∩ D (cid:48) contains a chamber. On the other hand, if M (cid:107) M (cid:48) then there exists D bounded by M and D (cid:48) bounded M (cid:48) such that M (cid:48) ⊂ D and M ⊂ D (cid:48) . Now we discuss residues in Σ , parallel to the discussion at the end of Subsection 5.2. Notation.
For I ⊂ S and g ∈ Γ , let g Σ I denote the union of the chambers of the I -residuecontaining gx . Notice that g Σ I = g Γ I x and Ch( g Σ I ) = g Γ I . For simplicity, in the following we alsocall a subset g Σ I ⊂ Σ a residue . Notice that a rotation around a building-wall that crosses g Σ I is of the form gγs α γ − g − with s ∈ I , s α (cid:54) = e and γ ∈ Γ I . By the definitions of theaction and the residues we obtain the following fact. Fact 5.24.
Let R = g Σ I be a residue. Then • R is stabilized by the rotations around the building-walls that cross it, • Stab Γ ( R ) = g Γ I g − is generated by these rotations, • The type I of R is equal to the set of types of all the building-walls M satisfying M crosses R and M is along gx . The following result gives a converse to the preceding fact. We recall that a set ofchambers is convex if it is convex for the combinatorial metric over the chambers (seeDefinition 5.3).
Theorem 5.25.
Let C ⊂ Ch(Σ) be a convex set of chambers. Let R = (cid:83) x ∈ C x ⊂ Σ andlet P R denote the group generated by the rotations around the building-walls that cross R .If P R stabilizes R , then R is a residue in Σ .Proof. Up to a translation on R and a conjugation on P R , we can assume that x ⊂ R . Westart by proving that P R acts freely and transitively on the set of chambers C . The actionof Γ is free thus the action of P R is free. For x ∈ C , by convexity of C , there exists a gallery x ∼ x ∼ · · · ∼ x (cid:96) = x of distinct chambers in C . Let M i be the building-wall of type s i ∈ S between x i − and x i .Then s α x = x , s α s α x = x , . . . , s α . . . s α (cid:96) (cid:96) x = x, α i ∈ Z .We notice that s α . . . s α i − i − s i ( s α . . . s α i − i − ) − is a rotation around M i . Therefore x maybe obtained from x by successive rotations around the building-walls M i . These building-walls cross R , thus the action is transitive. This proves that R = P R x and Stab Γ ( R ) = P R .It remains to prove that P R is of the form Γ I for a certain I ⊂ S .We set I ⊂ S the set of all the types of the building-walls that cross R along x and weidentify P R with Γ I . The inclusion Γ I < P R comes from the definitions of P R and I . Weproceed by induction on d c ( x , gx ) = (cid:96) to check that every element g of P R is a productof elements of Γ I . If (cid:96) = 0 , there is nothing to say. If (cid:96) > we choose g = s α . . . s α (cid:96) (cid:96) suchthat d c ( x , gx ) = (cid:96) . By convexity, s α x ∈ C so s ∈ Γ I . Indeed s is a rotation around abuilding-wall that crosses R along x . Then d c ( x , s − α gx ) = (cid:96) − and s − α g ∈ P R . Theinduction assumption allows us to conclude.In particular, this last theorem is used in Subsection 6.1 when we discuss the boundariesof the residues in the hyperbolic case. Finally we recall that the intersections of parabolicsubgroups in Γ (resp. in W ) is again a parabolic subgroup. Σ as a metric space A natural geodesic metric on Σ is obtained as follows. We designate by D the Davis chamberof Γ . We recall that D is obtained from D (1) by attaching a k -cube to every subgraph thatis isomorphic to the -skeleton of a k -cube. Now, for any k , we equip each k -cube of D with the Euclidean metric of the [0 , k .The polyhedral metric d ( · , · ) induced on Σ by this construction is geodesic and com-plete. Moreover, any automorphism of ∆ is an isometry of (Σ , d ) . In particular, Γ actsgeometrically on (Σ , d ) and, as Γ is assumed to be hyperbolic, (Σ , d ) is a hyperbolic metricspace.In (Σ , d ) the building-walls are convex and connected subsets and we can precise thisdescriptions using geodesic rays. Let M ∈ M (Σ) be of type s , let x ∈ Ch(Σ) such that M is along x and let σ s be the maximal face of type s of x . We denote by Ext( σ s ) the set of all the geodesic rays such that there exists (cid:15) > with r ([0 , (cid:15) )) ⊂ σ s . Then M = { r ( t ) : r ∈ Ext( σ s ) and t ∈ [0 , + ∞ ) } .However in the case when W is a reflection group of the hyperbolic space H d it seemsmore natural to equip D with the hyperbolic metric. Then D is isometric to the Cox-eter polytope provided by W . We designate by d (cid:48) ( · , · ) the piecewise hyperbolic metric on Σ induced by this construction. This metric satisfies the same properties stated above(geodesic, complete, hyperbolic and admits a geometric action of Γ ). The two metrics (Σ , d ) and (Σ , d (cid:48) ) are quasi-isometric. Since our goal is to study ∂ Γ , it makes no differenceto consider (Σ , d ) or (Σ , d (cid:48) ) . However, the arguments presented in Sections 6, 7, 8, and9 hold in the generic case so we consider (Σ , d ) in those sections. For Section 10, whichfocuses on hyperbolic reflection groups, it will be more convenient to consider (Σ , d (cid:48) ) .31 .10 Boundary of the building In this subsection we describe some basic properties of ∂ Γ . In the sequel, we use thegeometric action of Γ on (Σ , d ) to identify ∂ Γ and ∂ Σ . Now consider a building-wall M oftype s . We have ∂M (cid:39) ∂ Stab Γ ( M ) using the previous identification. Consider a subgroup P < Γ , as Γ is hyperbolic, either ∂P = ∂ Γ , or ∂P is of empty interior. Hence, here thereare two possible cases: • ∂M (cid:39) ∂ Γ , • Int( ∂M ) is empty.Moreover, the hyperbolicty assumption implies the following lemma. Lemma 5.26.
Let M and M (cid:48) be two distinct building-walls. If M (cid:107) M (cid:48) then ∂M ∩ ∂M (cid:48) = ∅ .Proof. Let γ : [0 , + ∞ ) −→ Σ be a geodesic ray contained in M . For simplicity we denoteby γ the image of γ . Assume that there exists K > such that dist( γ ( t ) , M (cid:48) ) ≤ K forevery t ≥ . Since M ∩ M (cid:48) = ∅ and because of the chamber structure, there exists K (cid:48) > such that K (cid:48) ≤ dist( γ ( t ) , M (cid:48) ) for every t ≥ .Now let Γ (cid:48) be the group generated by a rotation r around M and a rotation r (cid:48) around M (cid:48) . If the rotations are of order two, then Γ (cid:48) is an infinite dihedral group and the subset Γ (cid:48) γ ⊂ Σ is quasi-isometric to a Euclidean half space. If the rotations are of order largerthan two then Γ (cid:48) γ contains a proper subset quasi-isometric to a Euclidean half space.Now we can precise the description of the two cases above. In the first case, s commuteswith every generator r ∈ S . In the Davis complex this means that all the other building-walls are orthogonal to M . Then Stab Γ ( M ) = Γ .In the second case, there exists r ∈ S that does not commute with s . In the Davis com-plex this means that there exists a building-wall M (cid:48) parallel to M . This implies that ∂M (cid:32) ∂ Γ . In this case, ∂ Γ \ ∂M is the disjoint union Int( ∂D ( M )) (cid:116) · · · (cid:116) Int( ∂D q − ( M )) where D ( M ) , . . . , D q − ( M ) are the dials of building bounded by M . Naturally a rotation around M extends to the boundary as an homeomorphism that permutes ∂D ( M ) , . . . , ∂D q − ( M ) and fixes ∂M . Moreover Int( ∂M ) = ∅ , Int( ∂D i ( M )) (cid:54) = ∅ , and the topological boundaryof ∂D i ( M ) in ∂ Γ is ∂M for any i = 0 , . . . , q − . Concerning the dials of building, thefollowing alternative holds. Let D be a dial of building bounded by the building-wall M ,then: • either ∂M = ∂D = ∂ Γ , • or the topological boundary of ∂D in ∂ Γ is ∂M . In this case, Int( ∂D ( M )) (cid:54) = ∅ and Int( ∂M ) = ∅ .In [BK13, Proposition 5.2.], it is proved that the boundaries of half-spaces in a hyperbolicCoxeter group form a basis for the visual topology on the visual boundary. In the case ofright-angled building the analogous statement holds.32 act 5.27. The topology generated by { ∂D : D ∈ D (Σ) } is equivalent to the topologyinduced by a visual metric on ∂ Γ . Finally, consider an apartment A containing the base chamber x and the retractionmap π A,x : Σ −→ A . This retraction maps any geodesic ray of Σ starting from a basedpoint p ∈ x to a geodesic ray in A starting from p . Hence π A,x extends naturally to theboundaries and we keep the notation π A,x : ∂ Σ −→ ∂A for this extension. Remark 5.28.
In [DM02], M.W. Davis and J. Meier described how properties of connect-edness of ∂ Γ are encoded in the combinatorial structure of G . We use a corollary of theirresult in Subsection 6.2. Remark 5.29.
A classification of F. Haglund and F. Paulin states that the constructionpresented in Subsection 5.5 describes all the right-angled buildings in the following sense.
Theorem 5.30 ([HP03, Proposition 5.1.]) . Let Γ be the graph product given by the pair ( G , { G i } i =1 ,...,n ) as in Definition 5.13. Let ∆ be the building of type ( W, S ) associated with Γ . Assume that ∆ (cid:48) is a building of type ( W, S ) such that for any s i ∈ S the { s i } -residuesof ∆ (cid:48) are of cardinality G i . Then ∆ and ∆ (cid:48) are isomorphic. As we will see with Example 6.8, parabolic limit sets ( i.e boundaries of residues of thebuilding) play a key role in the proof of the CLP on boundaries of graph product.In this section, we use the ideas of [BK13, Section 5 and 6] to prove Theorem 6.12 whichis the first major step to prove the main result of this article (Theorem 10.1). The idea ofthis theorem is to control the modulus of the curves of a parabolic limit set by the modulusof curves in the neighborhood of a single curve. Then we apply this theorem to recover aresult about boundaries of right-angled Fuchsian buildings.We use the notation and convention of Section 5. In particular Γ is a fixed graph productgiven by the pair ( G , { Z /q i Z } i =1 ,...,n ) . We identify the building ∆ with its Davis complex Σ equipped with the piecewise Euclidean metric. The base chamber is x . We assume that Γ and Σ are hyperbolic and that ∂ Γ is arcwise connected. The metric on Ch(Σ) is denotedby d c ( · , · ) . Moreover, in this section we equip ∂ Γ with a self-similar metric that comes fromthe action of Γ on Σ . ∂ Γ In this subsection we give some basic properties of boundaries of parabolic subgroups. Atthe end of this subsection we will see that these subsets of the boundary cause difficultiesin proving the CLP. 33 efinition 6.1.
Let P = g Γ I g − be a parabolic subgroup of Γ . If the limit set of P in ∂ Γ is non-empty, we call it a parabolic limit set . If moreover ∂P (cid:54) = ∂ Γ the parabolic limit setis called a proper parabolic limit set. Equivalently we could say that a subset F ⊂ ∂ Γ is a parabolic limit set if there exists aresidue g Σ I such that F is equal to ∂ ( g Σ I ) under the canonical homeomorphism between ∂ Γ and ∂ Σ . In the following we will frequently use this point of view about parabolic limitsets.The following notion of convex hull of a subset of the boundary will be used in thissection. Definition 6.2.
Let F be a subset of ∂ Γ containing more than one point and such that F (cid:54) = ∂ Γ . Let D c ( F ) = { D ∈ D (Σ) : F ⊂ ∂ Γ \ ∂D } . The convex hull of F in Σ is defined by Conv( F ) = Σ \ ∪ D ∈D c ( F ) D. If F = ∂ Γ then we set Conv( F ) = Σ . Clearly we can also write
Conv( F ) = (cid:84) D ∈D c ( F ) Σ \ D . Hence Conv( F ) is convex for boththe geodesic metric on Σ and the combinatorial metric over the chambers of Σ . Moreoverwe observe that F ⊂ ∂ Conv( F ) . Example 6.3.
Let ∂P be a parabolic limit set and assume that P = Γ I . Then we can verifythat Conv( ∂P ) = Σ J where J = I ∪ { s j ∈ S : s j s i = s i s j for any s i ∈ I } .In particular, if M ∈ M (Σ) then Conv( ∂M ) is the the union of all the chambers along M . In the following definition
Σ = Σ ∪ ∂ Σ and if M is a building-wall M = M ∪ ∂M . Definition 6.4. i) Let F be a subset of ∂ Σ . We say that a building-wall M cuts F if there exist twodistinct indices i and j such that F meets both ∂D i ( M ) and ∂D j ( M ) .ii) If E ⊂ ∂ Σ and E ⊂ Σ (resp. E ⊂ ∂ Σ ) we say that a building-wall M separates E and E if E and E are entirely contained in two distinct connected components of Σ \ M . The proof of the following fact is identical to the proof of [BK13, Lemma 5.7].
Fact 6.5.
Let F be a subset of ∂ Σ . The building-wall M cuts F if and only if M crosses Conv( F ) (see Definition 5.22). Corollary 6.6.
Let F be a subset of ∂ Σ containing at least two distinct points and P F denote the group generated by the rotations around the building-walls that cut F . If P F stabilizes F , then F is a parabolic limit set. This characterization yields the following corollary concerning the connectedness of theparabolic limit sets.
Corollary 6.7.
Let ∂P be a parabolic limit set. Then every connected component F of ∂P containing more than one point is a parabolic limit set.Proof. Let M be a building-wall that cuts F . Since M cuts ∂P a rotation r ∈ Γ around M stabilize ∂P so in particular it permutes the connected components of ∂P . With r ( ∂M ∩ F ) = ∂M ∩ F we deduce that r ( F ) = F and so F is a parabolic limit set.Finally the following example illustrates the difficulty caused by parabolic limit sets inproving the CLP. Example 6.8.
Let M ∈ M (Σ) be a building-wall of type s along the base chamber x . Let P = Stab Γ ( M ) . The group P is the parabolic subgroup which is generated by the generators r ∈ S \{ s } such that rs = sr . Moreover, as we recalled in Subsection 5.10, ∂P (cid:39) ∂M . Nowwe assume that ∂P is a proper parabolic limit set and we pick g ∈ Γ . Now we verify that • either g∂P = ∂P , • or ∂P ∩ g∂P = ∅ .Indeed if two building-walls M and M are distinct with M ⊥ M then M and M areof distinct types. As M and gM are of the same type it follows that if M ∩ gM (cid:54) = ∅ then M = gM and g∂P = ∂P . On the other hand, if M ∩ gM = ∅ , by Lemma 5.26 thehyperbolicty implies that ∂P ∩ g∂P = ∅ .Finally the set ∪ g ∈ Γ g∂M is the union of countably many disjoint copies of ∂M . In theintroduction we recalled that an efficient way to prove the CLP is to follow curves using bi-Lipschitz maps (see Section 4). As Γ acts by bi-Lipschitz homeomorphisms on its boundary,the first idea is to use Γ to follow curves. However if a non-constant curve η is containedin ∂M then we cannot hope to follow the curves of ∂ Γ using η and Γ . In this subsection we apply the ideas of [BK13, Section 5 and 6] to Γ . As in Subsection 3.2, d denotes a small constant compared with diam ∂ Γ and with the constant of approximateself-similarity. Then F is the set of curves of diameter larger than d . Here we prove that,from the point of view of the modulus, curves in a parabolic limit set are all the same (seeconsequences of Theorem 6.12).Until the end of this article, we use the following notation: Notation.
Let ∂P be a connected parabolic limit set in ∂ Γ . For δ, r > , let F δ,r ( ∂P ) denote the set of curves γ in ∂ Γ such that: • diam γ ≥ d , • γ ⊂ N δ ( ∂P ) , • γ (cid:54)⊂ N r ( ∂Q ) for any connected parabolic limit set ∂Q (cid:32) ∂P . As we saw in Example 6.8, it is impossible to use the curves in the parabolic limit set tofollow other curves. Nevertheless, in this section we prove that these curves can be used tofollow the curves in ∂P (Proposition 6.11). Then we deduce from this proposition a controlof the modulus of the curves in parabolic limit sets (Theorem 6.12). To this end we use thefollowing notion. Definition 6.9.
Let L ≥ and I a non-empty subset of S . A curve γ in ∂ Γ is called a ( L, I ) -curve if • x ⊂ Conv( γ ) , • for all s ∈ I , there exists a panel σ s of type s inside Conv( γ ) with dist( x , σ s ) ≤ L . ( L, I ) -curves. Proposition 6.10.
Let I ⊂ S and P = h Γ I h − . Then for all r > , there exist L > and δ > such that if x ⊂ Conv( γ ) and γ ∈ F δ,r ( ∂P ) , then γ is a ( L, I ) -curve.Proof. We fix r > and we assume that for every integer n ≥ , there exists a curve γ n such that: • x ⊂ Conv( γ n ) , • γ n ∈ F /n,r ( ∂P ) , • γ n is not a ( n, I ) -curve.For n ≥ , we designate the ball of center x and of radius n for the metric over thechambers by B c ( x , n ) = { x ∈ Ch(Σ) : d c ( x , x ) ≤ n } . For simplicity we also designate by B c ( x , n ) the union of its chambers. Up to a subsequencewe can suppose that for a fixed s ∈ I , there is no panel of type s in B c ( x , n ) ∩ Conv( γ n ) for n ≥ .We want to reveal a contradiction using the sequences { γ n } n ≥ and { Conv( γ n ) } n ≥ .According to [Mun75, p. 281] the set of non-degenerate continua in a compact metricspace is a compact metric space with respect to the Hausdorff distance. Therefore, up to asubsequence, we can suppose that γ n tends to a non-degenerate continuum L ⊂ ∂P .Since x ⊂ Conv( γ n ) , using a diagonal argument we can also assume that, up to asubsequence, Conv( γ n + k ) ∩ B c ( x , n ) is non-empty and constant for k ≥ . We denote C := (cid:83) n ≥ Conv( γ n ) ∩ B c ( x , n ) . As we remarked before, if F ⊂ ∂ Γ contains more thanone point F ⊂ ∂ Conv( F ) . In particular L ⊂ ∂C .Now, by the assumptions on the sequence { γ n } n ≥ we observe that C does not containany panel of type s . Hence L is contained in the limit set of a parabolic subgroup of theform g Γ J g − with s / ∈ J . Moreover, we know that intersections of parabolic subgroups areparabolic subgroups (see [Cla13, Theorem 1.2]). Therefore L is contained in the limit setof the parabolic subgroup Q (cid:48) = P ∩ g Γ J g − . Let ∂Q be the connected component of ∂Q (cid:48) that contains L . Thanks to Corollary 6.7, ∂Q is a parabolic limit set. Now, as γ n tendsto L ⊂ ∂Q with respect to the Hausdorff distance, we have that for n ≥ large enough γ n ⊂ N r ( ∂Q ) .Finally, if ∂Q (cid:32) ∂P , this reveals a contradiction with γ n ∈ F /n,r ( ∂P ) for every n ≥ .If ∂Q = ∂P , then we apply the same reasoning with s (cid:48) ∈ I \{ s } .An interesting feature of ( L, I ) -curves is that these curves are crossed by building-wallsof type in I . Which means that from a ( L, I ) -curve, we can follow curves using rotationsaround building-walls of type in I . 37 roposition 6.11. Let (cid:15) > , L > and I be a non-empty subset of S . For P = h Γ I h − ,let η denote a curve contained in ∂P . Then there exists a finite subset F ⊂ Γ such that forany ( L, I ) -curve γ the set (cid:83) g ∈ F gγ contains a curve that belongs to U (cid:15) ( η ) .Proof. We divide the proof in four steps. In this proof M s denotes the building-wall of type s ∈ S along x .i) First, we can suppose without loss of generality that P = Γ I . Indeed, as h ∈ Γ is a bi-Lipschitz homeomorphism of ( ∂ Γ , d ) , then if the property holds for Γ I it holds for h Γ I h − .ii) Now we prove that the following property holds. The set (cid:83) g ∈ F L gγ contains a curve passing through ∂M s for every s ∈ I , where F L = { g ∈ Γ : | g | ≤ L } and | g | = d c ( x , gx ) .As γ is a ( L, I ) -curve, for any s ∈ I there exist α ∈ Z and g ∈ F L such that gx and gs α x belong to Conv( γ ) . We fix s ∈ I , α ∈ Z and g ∈ F L as before and let x ∼ g x ∼ · · · ∼ g (cid:96) x be a gallery contained in Conv( γ ) with g (cid:96) − = gs α and g (cid:96) = g . For any i = 0 , . . . , (cid:96) − , let M i denote the building-wall separating g i x and g i +1 x . In particular, ∂M i cuts γ for any i = 0 , . . . , (cid:96) − . This means that if M i is of type s i , then γ ∩ g i s α i i g − i γ (cid:54) = ∅ for any α i ∈ Z .In particular, if α i is such that g i +1 = g i s − α i i then γ ∩ g i g − i +1 γ (cid:54) = ∅ for any i = 0 , . . . , (cid:96) − .Hence the set γ ∪ g − γ ∪ · · · ∪ g − (cid:96) γ is arcwise connected and g − (cid:96) γ intersects ∂M s . Thusthe property is satisfied.iii) Let Σ I ⊂ Σ be the residue associated with Γ I . We recall that this means Σ I = Γ I x .For each ≤ i ≤ k let D i be a dial of building bounded by the building-wall M i . We assumethat each D i intersects Σ I properly ( i.e. Σ I ∩ D i (cid:54) = ∅ and Σ I ∩ D i (cid:54) = Σ I ). In particular,this means that the building-walls M , . . . , M k have their types contained in I .Now we prove that the following property holds. There exists a finite subset F ⊂ Γ such that for every ( L, I ) -curve γ theset (cid:83) g ∈ F gγ contains a curve passing through ∂D , . . . , ∂D k . For i = 1 , . . . , k pick h i ∈ Γ I such that M i is along h i x ∈ Σ I . In particular, for any i , wecan write M i = h i ( M s ) where s ∈ I is the type of M i . Let g x = h x ∼ g x ∼ · · · ∼ g (cid:96) x = h k x be a gallery in Σ I passing through h x , . . . , h k x in this order.Applying the second step of the proof, there exists a curve θ in (cid:83) g ∈ F L gγ such that θ crosses ∂M s for every s ∈ I . Therefore the set (cid:83) i =1 ,...,(cid:96) g i θ meets g i ( M s ) for any i = 1 , . . . , k and any s ∈ I . In particular, it meets every h i ( M s ) and intersects every ∂D , . . . , ∂D k .We set F = { g i g ∈ Γ : | g | ≤ L, ≤ i ≤ (cid:96) } , and it is now enough to check that (cid:83) g ∈ F gγ is arcwise connected. For any i = 1 , . . . , (cid:96) − let s i ∈ I and α i ∈ Z be such that g i +1 = g i s α i i .Then g i +1 θ = ( g i s α i i g − i ) g i θ . Since θ intersects any ∂M s i then g i θ ∩ g i ( ∂M s i ) (cid:54) = ∅ and thisintersection is fixed by g i s α i i g − i . Thus g i θ ∩ g i +1 θ (cid:54) = ∅ .iv) With Fact 5.27, we can choose, D (cid:48) , . . . , D (cid:48) k +1 a collection of dials of building suchthat the union of their boundaries is a neighborhood of η contained in the (cid:15)/ neighborhood38f η . We also assume that η enters in the boundaries of the D (cid:48) , . . . , D (cid:48) k +1 in this order.For any i = 1 , . . . , k + 1 , let r i denote the rotation around the building-wall associatedwith D (cid:48) i . Let D , . . . , D k be a collection of dials of building such that ∂D i ⊂ ∂D (cid:48) i ∩ ∂D (cid:48) i +1 .Applying the previous step of the proof, there exists a finite set F ⊂ Γ such that for every ( L, I ) -curve γ the set (cid:83) g ∈ F gγ contains a curve passing through each ∂D , . . . , ∂D k .If for some i = 1 , . . . , k + 1 the curve η leaves ∂D (cid:48) i then θ (cid:83) α ∈ Z r αi θ contains a curvethat does not leave ∂D (cid:48) i . Finally we set F = { r αi g : α ∈ Z and g ∈ F } and F satisfies thedesired property.We use the two preceding propositions to obtain a control of Mod p ( F δ,r ( ∂P )) . Theorem 6.12.
There exists an increasing function δ : (0 , + ∞ ) −→ (0 , + ∞ ) satisfyingthe following property. Let p ≥ , let η ∈ F , and let ∂P be the smallest parabolic limit setcontaining η . Let r > be small enough so that η (cid:54)⊂ N r ( ∂Q ) for any connected paraboliclimit set ∂Q (cid:32) ∂P and let δ < δ ( r ) . Then for (cid:15) > small enough there exists a constant C = C ( d , p, η, r, (cid:15) ) such that for every k ≥ C − · Mod p ( U (cid:15) ( η ) , G k ) ≤ Mod p ( F δ,r ( ∂P ) , G k ) ≤ C · Mod p ( U (cid:15) ( η ) , G k ) . Furthermore when p belongs to a compact subset of [1 , + ∞ ) the constant C may be chosenindependent of p .Proof. i) We can assume, without loss of generality, that x ⊂ Conv( γ ) for every γ ∈ F .Indeed, there exists an upper bound N depending on d such that dist( x , Conv( γ )) ≤ N for every γ ∈ F . So there exists only a finite set E of elements of Γ such that for g ∈ E ,there exists γ ∈ F with dist( x , Conv( γ )) = d c ( x , gx ) . Now, for F ⊂ F , we write F x = { γ ∈ F : x ⊂ Conv( γ ) } . Applying Proposition 3.11 N times and Proposition 2.3, we obtain that for every k ≥ large enough Mod p ( F x , G k ) ≤ Mod p ( F , G k ) ≤ C · Mod p ( F x , G k ) , where C depends only on d .ii) Similarly, for (cid:15) > small enough, we can assume that U (cid:15) ( η ) ⊂ F . Indeed, for (cid:15) > small enough there exists an upper bound N depending on d such that if γ ∈ U (cid:15) ( η ) then dist( x , Conv( γ )) ≤ N . Again, the multiplicative constant induced by this assumptiondepends only on d .iii) As a consequence of the preceding part, for (cid:15) > small enough, one has U (cid:15) ( η ) ⊂F δ,r ( ∂P ) and the left-hand side inequality is established by Proposition 2.3 (1).iv) Now we prove the right-hand side inequality. Let P = h Γ I h − , let η be a curve in ∂P , r > as in the hypothesis of the theorem and (cid:15) > small enough so that the precedingpart hold. 39ith the assumption of the first part, we can apply Proposition 6.10 and set L > and δ > such that the curves of F δ,r ( ∂P ) are ( L, I ) -curves. Let F ⊂ Γ be the finite set givenby Proposition 6.11 and let ρ : G k −→ [0 , + ∞ ) be a U (cid:15) ( η ) -admissible function. We define ρ (cid:48) : G k −→ [0 , + ∞ ) by: ( ∗ ) ρ (cid:48) ( v ) = (cid:88) g ∈ F (cid:88) w ∩ gv (cid:54) = ∅ ρ ( w ) . Let γ ∈ F δ,r ( ∂P ) and θ ⊂ (cid:83) g ∈ F gγ such that θ ∈ U (cid:15) ( η ) . Then L ρ (cid:48) ( γ ) = (cid:88) g ∈ F (cid:88) v ∩ γ (cid:54) = ∅ (cid:88) w ∩ gv (cid:54) = ∅ ρ ( w ) ≥ (cid:88) g ∈ F (cid:88) w ∩ gγ (cid:54) = ∅ ρ ( w ) . However L ρ ( θ ) ≤ (cid:88) g ∈ F L ρ ( gγ ) = (cid:88) g ∈ F (cid:88) w ∩ gγ (cid:54) = ∅ ρ ( w ) . Thus L ρ (cid:48) ( γ ) ≥ L ρ ( θ ) and ρ (cid:48) is F δ,r ( ∂P ) -admissible.Then the number or terms in the right-hand side of the definition ( ∗ ) is bounded bya constant N depending on F , the bi-Lipschitz constants of the elements of F , and thedoubling constant of ∂ Γ . Therefore by convexity M p ( ρ (cid:48) ) = (cid:88) v ∈ G k (cid:16) (cid:88) g ∈ F (cid:88) w ∩ gv (cid:54) = ∅ ρ ( w ) (cid:17) p ≤ N p − · (cid:88) v ∈ G k (cid:88) g ∈ F (cid:88) w ∩ gv (cid:54) = ∅ ρ ( w ) p ≤ N p · F · (cid:88) w ∈ G k ρ ( w ) p . Which proves the inequality. The multiplicative constant induced by this part depends on p , η and r as the set F depends on η and r . However it does not depend on k .v) Here we write δ ( r, ∂P ) for the constant δ chosen in the previous part such that thecurves of F δ,r ( ∂P ) are ( L, I ) -curves for a certain L > .By Proposition 7.20 there exist only a finite number of parabolic limit set of diameterlarger than d . Moreover we recall that according to our notation, if δ ≥ δ (cid:48) then F δ (cid:48) ,r ( ∂P ) ⊂F δ,r ( ∂P ) . As a consequence, the function defined by δ ( r ) = min { δ ( r, ∂P ) : diam ∂P ≥ d } , satisfies the desired property.vi) As we see at the end of part iv), C = λ.N p with λ and N independent of p . Henceif p belongs to a compact subset K ⊂ [1 , + ∞ ) , the constant C may be chosen independentof p by taking C = λ.N max K .As an immediate application, we notice that under the assumptions of the theorem, thebehavior of Mod p ( U (cid:15) ( η ) , G k ) as k goes to infinity does not depend, on the choice of η and40 . Indeed, for p ≥ , r > and δ < δ ( r ) fixed, if η, η (cid:48) ⊂ ∂P and (cid:15), (cid:15) (cid:48) > are such that thehypothesis of the theorem are satisfied. Then there exist C = C ( η, (cid:15) ) and C (cid:48) = C (cid:48) ( η (cid:48) , (cid:15) (cid:48) ) such that C − · Mod p ( U (cid:15) (cid:48) ( η (cid:48) ) , G k ) ≤ Mod p ( U (cid:15) ( η ) , G k ) ≤ C (cid:48) · Mod p ( U (cid:15) (cid:48) ( η (cid:48) ) , G k ) . Of course, if η = η (cid:48) and (cid:15) (cid:48) < (cid:15) we can choose C = 1 . Another consequence of the theorem is that if the boundary of a graph product does notcontain connected parabolic limit sets, then it satisfies the CLP.
Theorem 6.13.
Let Γ be a thick hyperbolic graph product such that ∂ Γ is connected and anyproper parabolic limit set is disconnected. Then ∂ Γ equipped with a visual metric satisfiesthe CLP.Proof. We check the hypothesis of Proposition 3.12. To prove that
Mod ( F , G k ) is un-bounded, it is enough to check that for every N ∈ N there exist N disjoint curves of diameterlarger than d in ∂ Γ . Indeed, this implies that for k ≥ large enough Mod ( F , G k ) > N .To prove this we use the assumption on the thickness which implies that for every N ∈ N there exist N apartments with disjoint boundaries that intersects in a compactdomain inside the building. To observe such apartments we use the following notation. • W i − = { w ∈ Γ : w = s i . . . s ik with s j ∈ S and d c ( x , wx ) = k } for i = 1 , . • CB n = Conv( B c ( x , n )) ⊂ Ch(Σ) the convex hull for the metric over the chambers d c ( · , · ) of B c ( x , n ) ⊂ Ch(Σ) the ball of center x and of radius n for d c ( · , · ) for n ≥ . • FCB n is the frontier of CB n . By this we mean the set of all the chambers x ∈ CB n such that there exists y / ∈ CB n with x ∼ y .Then the following sets of chambers define N apartments all containing the base chamber x , and intersecting only inside CB N • A = { wx : w ∈ W } , • A = { wx : w ∈ W } , • A = A ∩ CB (cid:83) { gwx : w ∈ W , gx ∈ A ∩ FCB } , • A n = A n − ∩ CB n − (cid:83) { gwx : w ∈ W n +1 mod(2) and gx ∈ A n − ∩ FCB n − } for n = 2 , . . . , N . 41ow let η be a non-constant curve in ∂ Γ . Up to a change of scale, by Proposition 3.11,we can assume η ∈ F . Then as ∂ Γ is the only parabolic limit set containing η , it is enoughto apply Theorem 6.12 to satisfy the second hypothesis of Proposition 3.12.In particular, we can apply this result to the case of right-angled Fuchsian buildings.In the following, we call right-angled Fuchsian building a building associated with a graphproduct ( C n , { Z /q i Z } i =1 ,...,n ) where C n is the cyclic graph with n ≥ vertices and q , . . . , q n is a family of integers larger than or equal to . Corollary 6.14.
For n ≥ , let C n be the cyclic graph with n vertices and let q , . . . , q n bea family of integers larger than or equal to . Let Γ be the graph product given by the pair ( C n , { Z /q i Z } i =1 ,...,n ) . Then ∂ Γ equipped with a visual metric satisfies the CLP. This result was known since boundaries of right-angled Fuchsian buildings are Loewnerspaces (see [BP00, Proposition 2.3.4.]). However, here we give a direct proof of this result.Furthermore, we can prove that these thick graph products are the only ones that satisfythe hypothesis of Theorem 6.13. To verify this we need to introduce the following simplicialcomplex.
Definition 6.15.
Let
Γ = ( G , { Z /q i Z } i =1 ,...,n ) be a graph product, the nerve of Γ is thesimplicial complex L = L ( G ) such that: • the 1-skeleton of L is G , • k vertices of G span a ( k − -simplex in L if and only if the corresponding parabolicsubgroup in Γ is finite. For a simplex σ ⊂ L spanned by the vertices v , . . . , v k ∈ G (0) , we denote by G σ thefull sub-graph of G spanned by the vertices G (0) \{ v , . . . , v k } . Now we write L \ σ = L ( G σ ) .Note that L \ σ can be seen as a subcomplex of L .The following theorem is a special case of [DM02, Corollary 5.14.]. Theorem 6.16.
The boundary of Γ is connected if and only if the subcomplex L \ σ isconnected for any simplex σ ⊂ L . Now we can prove the following.
Proposition 6.17.
Let
Γ = ( G , { Z /q i Z } i =1 ,...,n ) be a hyperbolic graph product. Assumethat ∂ Γ is connected and that any proper parabolic limit set ∂P is disconnected, then thebuilding associated with Γ is a right-angled Fuchsian building.Proof. We only need to prove that G contains a circuit of length n ≥ . According toCorollary 6.7, if any proper parabolic limit set in ∂ Γ is disconnected then any properparabolic limit set in ∂ Γ is discrete. Moreover, ∂ Γ contains at least one proper parabolic42imit set of the form ∂ Γ I with I = n − otherwise ∂ Γ = ∅ . The subgroup Γ I is a graphproduct associated with the graph G I . This graph is obtained from G to which we removea vertex p and all the edges adjacent to p . Then if L I is the nerve associated with Γ I , weget L I from L to which we remove the interior of any simplex containing p .Now, thanks to Theorem 6.16, we know that there exists a simplex σ ⊂ L I such that L I \ σ is disconnected. Let C and C be two connected components of L I \ σ . Up to asubsimplex, we can assume that any vertex of σ is connected to C or to C by an edge.However, if we consider the simplex σ in L , we see that L \ σ is connected because ∂ Γ isconnected. Therefore there exist at least one edge attaching p to C and at least one edgeattaching p to C .We set V = { v , . . . , v k } the vertices of σ that are not connected to p by an edge and V (cid:48) = { v (cid:48) , . . . , v (cid:48) k (cid:48) } the rest of the vertices of σ . At this point, we assume by contradiction,that G contains no circuit of length n ≥ . We can check that under this assumption thefollowing situations does not occur C C pσ Figure 3: Forbidden situation i ) C C pσ Figure 4: Forbidden situation ii ) C C pσ Figure 5: Forbidden situation iii ) C C pσ (cid:48) V Figure 6: Resulting situationi) V (cid:48) is empty,ii) there exists v ∈ V such that v is adjacent to both C and C ,iii) there exist v, w ∈ V such that v is adjacent to C and w is adjacent to C .Hence, the vertices in V are either all adjacent to C or all adjacent to C . Assumethat the vertices in V are all adjacent to C . As a consequence, if σ (cid:48) designates the simplexin L spanned by V (cid:48) ∪ { p } then L \ σ (cid:48) is not connected. This is not possible because ∂ Γ isconnected.Therefore G contains a circuit of length n ≥ , but as Γ is hyperbolic it contains nocircuit of length . This concludes the proof.43 Combinatorial metric on boundaries of right-angled hyper-bolic buildings
In this section we explain how the geometry of the boundary is determined by boundariesof building-walls. We start by discussing the geometry of intersections of dials of buildingand the boundaries of such intersections. Then, we describe a combinatorial and self-similarmetric on ∂ Γ in terms of dials of building. Finally, we construct an approximation of ∂ Γ that will be convenient to use in Section 8.Here we use the notation and assumptions of Section 5 and 6. In particular, Γ is afixed graph product given by the pair ( G , { Z /q i Z } i =1 ,...,n ) and acting on the building Σ .The base chamber is x , and W is the right-angled Coxeter group associated with Γ . Weassume that Γ is hyperbolic and ∂ Γ is connected. Σ In right-angled buildings, we can project chambers on residues and on dials of building.This will be useful in the rest of this section to understand the metric on the boundary inthe hyperbolic case.
Proposition 7.1.
Let D be a residue or a dial of building and C = Ch( D ) . Then for any x ∈ Ch(Σ) there exists a unique chamber proj C ( x ) ∈ C such that d c ( x, proj C ( x )) = dist( x, C ) . Moreover, for any chamber y ∈ C there exists a minimal gallery from x to y passing through proj C ( x ) . For simplicity, in the following we sometime use the notation proj D ( · ) to designate proj Ch( D ) ( · ) . Before the proof of the proposition we need to establish the following lemma. Lemma 7.2.
Let M and M (cid:48) be two building-walls such that M ⊥ M (cid:48) . Let r ∈ Γ be arotation around M and D (cid:48) be a dial of building bounded by M (cid:48) then r ( D (cid:48) ) = D (cid:48) .Proof. Up to a translation on the dials and a conjugation on the rotations we can assumethat M and M (cid:48) are along x and x ⊂ D (cid:48) . If s is a rotation around M (cid:48) , we can write Ch( D (cid:48) ) = { x ∈ Ch(Σ) : d c ( x , x ) < d c ( x , sx ) } . Hence r (Ch( D (cid:48) )) = { rx ∈ Ch(Σ) : d c ( x , x ) < d c ( x , sx ) } = { x ∈ Ch(Σ) : d c ( x , r − x ) < d c ( x , sr − x ) } . By assumption rs = sr , thus r (Ch( D (cid:48) )) = { x ∈ Ch(Σ) : d c ( rx , x ) < d c ( rx , sx ) } .Moreover, d c ( x , rx ) = 1 and d c ( x , srx ) = 2 so rx ∈ Ch( D (cid:48) ) . With Fact 5.20 we obtain Ch( D (cid:48) ) = r (Ch( D (cid:48) )) . 44 roof of Proposition 7.1. If D is a residue, then we refer to [Tit74, Proposition 3.19.3]. If D is a dial of building, let y ∈ C be such that d c ( x, y ) = dist( x, C ) . Then for z ∈ C we set x = x ∼ x ∼ · · · ∼ y and y = x (cid:96) ∼ · · · ∼ z = x k two minimal galleries. Assume that thegallery x = x ∼ x ∼ · · · ∼ y = x (cid:96) ∼ · · · ∼ z = x k is not minimal. Then there exists a building-wall M and two indices i, j with ≤ i < (cid:96) and (cid:96) ≤ j < k such that • M separates x i and x i +1 , • M separates x j and x j +1 .Now consider r ∈ Γ the rotation around M such that rx i +1 = x i . By Proposition 7.2, r ( D ) = D hence, the gallery x ∼ · · · ∼ x i ∼ rx i +2 · · · ∼ rx (cid:96) = ry connects x to C and is of length dist( x, C ) − , which is a contradiction. x x i = rx i +1 x i +1 ryy CM Figure 7We proved that for any z ∈ C , there exists a minimal gallery from x to z passing through y . This proves in particular that y is unique and the proof is achieved.The following lemma says that the projections on the dials of building are orthogonalrelatively to the building-wall structure. Lemma 7.3.
Let
D, D (cid:48) ∈ D (Σ) such that
Ch( D ∩ D (cid:48) ) (cid:54) = ∅ . If x ∈ Ch( D ) then proj D (cid:48) ( x ) ∈ Ch( D ∩ D (cid:48) ) .Proof. Clearly proj D (cid:48) ( x ) ⊂ D (cid:48) so we check that proj D (cid:48) ( x ) ⊂ D . Under the assumption Ch( D ∩ D (cid:48) ) (cid:54) = ∅ three cases are possible. First, if D (cid:48) ⊂ D then proj D (cid:48) ( x ) ⊂ D (cid:48) ⊂ D . Thenif D ⊂ D (cid:48) , then proj D (cid:48) ( x ) = x ⊂ D for any x ∈ Ch( D ) .45ow let M and M (cid:48) be the building-walls that bound D and D (cid:48) . The last case is realizedwhen M ⊥ M (cid:48) . In this case consider a minimal gallery x ∼ · · · ∼ proj D (cid:48) ( x ) .If proj D (cid:48) ( x ) (cid:54)⊂ D , then the preceding gallery crosses M . As a consequence, we can writethat there exists a minimal gallery of the form x ∼ · · · ∼ x i ∼ x i +1 = rx i ∼ x i +2 ∼ · · · ∼ proj D (cid:48) ( x ) where r ∈ Γ is rotation around M . Then with Lemma 7.2 we obtain that x ∼ · · · ∼ x i ∼ r − x i +2 ∼ · · · ∼ r − proj D (cid:48) ( x ) is a gallery between x and D (cid:48) of length d c ( x, proj D (cid:48) ( x )) − . Which is a contradiction.Applying several times the projection maps on dials of building, we define projectionmaps on finite intersections of dials of building. Proposition 7.4.
Let D , . . . , D k ∈ D (Σ) and C = Ch( D ∩· · ·∩ D k ) . Assume that C (cid:54) = ∅ .Then for any x ∈ Ch(Σ) there exists a unique chamber proj C ( x ) ∈ C such that d c ( x, proj C ( x )) = dist( x, C ) . Moreover, for any chamber y ∈ C there exists a minimal gallery from x to y passing through proj C ( x ) . Finally proj C ( x ) = proj D k ◦ · · · ◦ proj D ( x ) . For simplicity, in the following we will use the notation proj D ( · ) instead of proj C ( · ) .Notice that it is not always possible to define a projection on a convex set of chambers.For instance, if Σ is a thick building there exist pairs of adjacent chambers x and y with d c ( x , x ) = d c ( x , y ) . Proof.
First, according to Lemma 7.3, we can assume, up to a subfamily, that x (cid:54)⊂ D i foreach i = 1 , . . . , k . Now we set • C = Ch( D ) and C i = C i − ∩ Ch( D i ) for any i = 2 , . . . , k , • x = proj D ( x ) and x i = proj D i ( x i − ) for any i = 2 , . . . , k .By induction on i we prove the following property: x i ∈ C i and is the unique chamber of C i such that d c ( x, x i ) = dist( x, C i ) .Moreover, for any chamber y ∈ C i there exists a minimal gallery from x to y passing through x i . i = 1 the property holds by Proposition 7.1. Let i > and assume that the propertyholds at rank i . In particular, for j = 1 , . . . , i one has x i ∈ Ch( D j ) and Ch( D j ) ∩ Ch( D i +1 ) (cid:54) = ∅ . Therefore, with Lemma 7.3, x i +1 ∈ Ch( D ) ∩ · · · ∩ Ch( D i ) ∩ Ch( D i +1 ) = C i +1 .By Proposition 7.1, x i +1 is the unique chamber in C i +1 such that d c ( x i , x i +1 ) = dist( x i , C i +1 ) . Moreover, by the same proposition, for y ∈ C i if the galleries x ∼ · · · ∼ x i and x i ∼ · · · ∼ y are minimal, the gallery x ∼ · · · ∼ x i ∼ · · · ∼ y is minimal. As a consequence, x i +1 is the unique chamber in C i +1 such that d c ( x, x i +1 ) =dist( x, C i +1 ) . On the other hand, by Proposition 7.1, for any y ∈ C i +1 there exists a minimal gallery x i ∼ · · · ∼ x i +1 ∼ · · · ∼ y. Hence if the gallery x ∼ · · · ∼ x i is minimal, the gallery x ∼ · · · ∼ x i ∼ · · · ∼ x i +1 ∼ · · · ∼ y is also minimal. ∂ Γ The following notions are used in the rest of this article to describe the topology and themetric on ∂ Γ . We recall that the boundary of Γ is canonically identified with the boundaryof Σ . Definition 7.5.
Let x ∈ Ch(Σ) . We call cone of chambers of base x and we denote C x ⊂ Σ ,the union of the set of chambers y ∈ Ch(Σ) such that there exists a minimal gallery from x to y passing through x . Cones of chambers are characterized by projection maps and dials of building.
Proposition 7.6.
Let D , . . . , D k ∈ D (Σ) and C = D ∩ · · · ∩ D k . Assume that C containsa chamber and that x (cid:54)⊂ D i for i = 1 , . . . , k . If we set x = proj C ( x ) then C x = C .Proof. According to Definition 7.5 and to Proposition 7.4, C ⊂ C x . Now let y ∈ Ch( C x ) andlet M i be the building-wall that bounds D i for each i = 1 , . . . , k . If x ∼ · · · ∼ x ∼ · · · ∼ y is a minimal gallery, then the subgallery x ∼ · · · ∼ y does not cross the building-wall M i for any i = 1 , . . . , k and y ⊂ C .Reciprocally cones are intersections of dials of building.47 roposition 7.7. Let x ∈ Ch(Σ) and let D , . . . , D k ∈ D (Σ) be the family of dials ofbuilding such that for any i = 1 , . . . , kx (cid:54)⊂ D i and x ⊂ D i . Then C x = D ∩ · · · ∩ D k .Proof. Let C = D ∩ · · · ∩ D k . According to Proposition 7.6, it is enough to prove that proj C ( x ) = x . If we write x (cid:48) = proj C ( x ) , with Proposition 7.6, C = C x (cid:48) . Hence thereexists a minimal gallery x ∼ · · · ∼ x (cid:48) ∼ · · · ∼ x. Now assume that x (cid:48) (cid:54) = x , this means that there exists a building-wall M that separates x and x (cid:48) . Since the preceding gallery is minimal, the dial of building D bounded by M that contains x does not contain x (cid:48) and x . Thus D ∈ { D , . . . , D k } and x (cid:48) (cid:54)⊂ C which isa contradiction.In particular, as cones of chambers are intersections of dials of building, it makes sense toconsider projection maps on cones and proj C x ( x ) = x . In the following fact we summarizewhat we can say about intersections of dials of building. Fact 7.8.
Let D , . . . , D k ∈ D (Σ) be such that D i (cid:54)⊂ D j for any i (cid:54) = j . Assume that x (cid:54)⊂ D i for any i = 1 , . . . , k . Let M i be the building-wall that bounds D i for any i = 1 , . . . , k , andlet C = D ∩ · · · ∩ D k . Then exactly one of these assertions holds. • There exists i, j such that M i (cid:107) M j and C = ∅ . • M i ∩ M j (cid:54) = ∅ for any i, j = 1 , . . . , k , and there exists i (cid:54) = j such that M i = M j . Inthis case C is contained in M i . • M i ⊥ M j for any i (cid:54) = j . In this case C is a cone. This fact, up to a translation and up to a subfamily, describes how a finite family ofdials intersects. The following lemma specifies the case when the intersection is a cone.
Lemma 7.9.
Let D , . . . , D k be a family of distinct dials of building bounded by the building-walls M , . . . , M k . Let C = D ∩ · · · ∩ D k . Assume that x (cid:54)⊂ D i and that M i ⊥ M j for any i, j = 1 , . . . , k with i (cid:54) = j . Then any M i is along proj C ( x ) .Proof. First if k = 1 the property is clearly satisfied. According to Lemma 7.3, proj D ( x ) / ∈ Ch( D ) ∪ · · · ∪ Ch( D k ) . Applying this lemma k − times we obtain that proj D k − ◦ · · · ◦ proj D ( x ) / ∈ Ch( D k ) . Hence proj C ( x ) = proj D k (proj D k − ( ◦ ) · · · ◦ proj D ( x )) is along M k . Finally, we apply thesame argument to M , . . . , M k − and the proof is finished.48inally we obtain the following characterization of cones. Proposition 7.10.
Let x ∈ Ch(Σ) and let D , . . . , D k be the family of dials of buildingbounded by M , . . . , M k such that for any i = 1 , . . . , kx (cid:54)⊂ D i , x ⊂ D i , and M i is along x. Theni) C x = D ∩ · · · ∩ D k ,ii) there exist g ∈ Γ and M , . . . , M k ∈ M (Σ) with • for any i (cid:54) = j : M i ⊥ M j , • for any i : M i is along x ,such that C x = g ( C ) where C = D ( M (cid:48) ) ∩ · · · ∩ D ( M (cid:48) k ) .Proof. Let D (cid:48) , . . . , D (cid:48) (cid:96) be the family of dials of building such that x (cid:54)⊂ D i and x ⊂ D i forany i = 1 , . . . , (cid:96) . Then { D , . . . , D k } ⊂ { D (cid:48) , . . . , D (cid:48) (cid:96) } . According to Proposition 7.7, C x = D (cid:48) ∩ · · · ∩ D (cid:48) (cid:96) , thus C x ⊂ D ∩ · · · ∩ D k . Let M (cid:48) i be the wall that bounds D (cid:48) i for any i = 1 , . . . , (cid:96) . Up to a subfamily, we can assume that C x = D (cid:48) ∩ · · · ∩ D (cid:48) (cid:96) with M (cid:48) i ⊥ M (cid:48) j for any i (cid:54) = j . Then, as proj C x ( x ) = x , by Lemma 7.9,any building-wall M (cid:48) i is along x . Finally, we get { D (cid:48) , . . . , D (cid:48) (cid:96) } ⊂ { D , . . . , D k } and D ∩ · · · ∩ D k ⊂ C x .The second part is immediate with g ∈ Γ such that gx = x .The following lemma is used to prove that boundaries of cones are of non-empty interior. Lemma 7.11.
Let M , . . . , M k be a collection of building-walls such that any M i admits aparallel building-wall. Assume that M i ⊥ M j for any i (cid:54) = j . Then there exists M ∈ M (Σ) such that M (cid:107) M i for any i .Proof. We prove the proposition by induction on k . For k = 1 there is nothing to prove.For k ≥ , pick a collection of building-walls M , . . . , M k satisfying the hypothesis of thelemma. Assume that there exists M a building-wall such that M (cid:107) M , . . . , M (cid:107) M k . Let M k +1 ∈ M (Σ) be such that M k +1 ⊥ M , . . . , M k +1 ⊥ M k .49f M is parallel to M k +1 there is nothing more to say. Now we assume M ⊥ M k +1 andwe pick a wall M (cid:48) parallel to M k +1 . If M (cid:48) is parallel to M , . . . , M k there is nothing moreto say. Now we assume that there exists ≤ i ≤ k such that, up to a reordering M (cid:48) ⊥ M , . . . , M (cid:48) ⊥ M i , M (cid:48) (cid:107) M i +1 , . . . , M (cid:48) (cid:107) M k , M (cid:48) (cid:107) M k +1 . First we consider the case M (cid:48) ⊥ M . With • M ⊥ M k +1 , M k +1 ⊥ M , M ⊥ M (cid:48) , • and M (cid:48) (cid:107) M k +1 , M (cid:107) M ,we obtain that the building-walls M (cid:48) , M, M k +1 , and M form a right-angled rectangle.Which is a contradiction with the hyperbolicity of Σ .Secondly we consider the case M (cid:48) (cid:107) M . Let r (cid:48) ∈ Γ be a rotation around M (cid:48) . Then r (cid:48) ( M ) is such that r (cid:48) ( M ) (cid:107) M , . . . , r (cid:48) ( M ) (cid:107) M i . Indeed, as M (cid:107) M j , it comes that r (cid:48) ( M ) (cid:107) r (cid:48) ( M j ) and, as M (cid:48) ⊥ M j , by Lemma 7.2, r (cid:48) ( M j ) = M j for ≤ j ≤ i . Thus r (cid:48) ( M ) (cid:107) M j .Moreover r (cid:48) ( M ) (cid:107) M i +1 , . . . , r (cid:48) ( M ) (cid:107) M k +1 . Indeed, as M i +1 ∩ · · · ∩ M k +1 (cid:54) = ∅ and M (cid:48) (cid:107) M j for i + 1 ≤ j ≤ k + 1 , the building-walls M i +1 , . . . , M k +1 are entirely contained in the same connected component of Σ \ M (cid:48) . Let C be this connected component. Since M (cid:107) M (cid:48) and M ∩ M k +1 (cid:54) = ∅ it comes that M is alsocontained in C . Thus r (cid:48) ( M ) is not contained in C and r (cid:48) ( M ) (cid:107) M j for i + 1 ≤ j ≤ k + 1 . Proposition 7.12.
Let x ∈ Ch(Σ) and C x be the cone based at x . Then ∂C x is of non-empty interior in ∂ Γ .Proof. By Fact 7.8 and Proposition 7.10 we can write ∂C x = ∂D ∩ · · · ∩ ∂D k , where D , . . . , D k is a collection of dials of building bounded by the building-walls M , . . . , M k with M i ⊥ M j for any i (cid:54) = j .Up to a subfamily, we can assume that for every i = 1 , . . . , k there exists M (cid:48) i ∈ M (Σ) such that M i (cid:107) M (cid:48) i . Indeed if M admits no parallel building wall then ∂M = ∂ Γ (seebeginning of Section 5.10).By rotations around M , . . . , M k , all the connected components of Σ \ ( M ∪ · · · ∪ M k ) are isomorphic. Hence, thanks to Lemma 7.11, there exists M ∈ M (Σ) such that M (cid:107) M i for any i = 1 , . . . , k and M ⊂ D ∩ · · · ∩ D k . In particular, there exists D ∈ D (Σ) bounded by M such that D ⊂ D ∩ · · · ∩ D k . As ∂D is of non-empty interior, we obtain that ∂C x is of non-empty interior.50n the rest of this article, we use boundaries of cones as a base for the topology of ∂ Γ and to construct approximations. Definition/Notation 7.13.
Let x ∈ Ch(Σ) and C x be the corresponding cone of chambers.We call shadow of x the boundary of C x in ∂ Γ and we write v x = ∂C x . ∂ Γ Until now we have been considering on ∂ Γ the visual metric coming from the geometricaction of Γ on Σ . Now we use minimal galleries to describe a combinatorial metric on ∂ Γ that will be more convenient to use in the sequel. We extend the notion of minimal galleryto infinite galleries. Definition 7.14.
An infinite gallery x ∼ x ∼ · · · (resp. a bi-infinite gallery · · · ∼ x − ∼ x ∼ x ∼ · · · ) is minimal if for any k ∈ N (resp. k ∈ Z ) and (cid:96) ∈ N the gallery x k ∼ · · · ∼ x k + (cid:96) is minimal. Let DG (Σ) denote the dual graph of Σ . This graph is defined by: • The set of vertices DG (Σ) (0) is given by Ch(Σ) the set of chambers in Σ . If v ∈DG (Σ) (0) then c v denotes the corresponding chamber in Ch(Σ) . • There exists an edge between two vertices v and v if and only if c v is adjacent to c v in Σ . • Each edge is isometric to the segment [0 , .Naturally, DG (Σ) is a proper geodesic and hyperbolic space. It is quasi-isometric to Σ and the action of Γ on DG (Σ) is geometric. Therefore we identify ∂ DG (Σ) (cid:39) ∂ Γ . Example 7.15.
We recall that the group Γ is given by the following presentation Γ = (cid:104) s i ∈ S | s q i i = 1 , s i s j = s j s i if v i ∼ v j (cid:105) . If q i = 2 or for any i = 1 , . . . , n then DG (Σ) is identified with Cay(Γ) the Cayley graph of Γ with respect to the recalled generating set. Otherwise, if we consider a generator s ∈ S of Γ of order q ≥ then in DG (Σ) the full sub-graph generated by the vertices associated with e, s, . . . , s q − is a complete graph. In Cay(Γ) the full sub-graph generated by the verticesassociated with e, s, . . . , s q − is a cyclic graph of length q . Nevertheless DG (Σ) and Cay(Γ) are always quasi-isometric.
With Definition 7.14, infinite minimal galleries are identified with geodesic rays in DG (Σ) starting from a vertex. Therefore we can identify ∂ Γ with the set of equivalence classes ofinfinite galleries starting at x where two such galleries x ∼ x ∼ · · · and y = x ∼ y ∼· · · are equivalent if and only if there exists K > such that d c ( x i , y i ) < K for all i ∈ N .51 xample 7.16. Here we consider only minimal galleries. We write R the equivalencerelation on the infinite galleries starting at x defined above. Let x ∈ Ch(Σ) with d c ( x , x ) = k ≥ . Then we can describe the shadow v x as follows v x (cid:39) { x ∼ x ∼ · · · ∼ x i ∼ · · · : x i ∈ Ch(Σ) and x k = x } / R . Likewise, if ∂P is a parabolic limit set associated with the residue g Σ I . Let x := proj g Σ I ( x ) and assume that d c ( x , x ) = k ≥ . Then we can describe ∂P as follows ∂P (cid:39) { x ∼ x ∼ · · · : x k = x and x k + i ∼ s i x k + i +1 with s i ∈ I for any i ≥ } / R . Now we use the following notation.
Notation. If x ∼ x ∼ · · · is a minimal infinite gallery that goes asymptotically to ξ ∈ ∂ Γ ,then we write ξ = [ x ∼ x ∼ · · · ] . Definition 7.17.
Let ξ, ξ (cid:48) be two distinct points in ∂ Γ , let { ξ | ξ (cid:48) } x denote the largest integer (cid:96) such that there exist two infinite minimal galleries representing ξ and ξ (cid:48) ξ = [ x ∼ x ∼ · · · ∼ x i ∼ · · · ] and ξ (cid:48) = [ x ∼ x (cid:48) ∼ · · · ∼ x (cid:48) i ∼ · · · ] with x i = x (cid:48) i for i ≤ (cid:96) and x (cid:96) +1 (cid:54) = x (cid:48) (cid:96) +1 . In terms of shadows, { ξ | ξ (cid:48) } x is the largest integer such that there exists a shadow v x ,with d c ( x , x ) = { ξ | ξ (cid:48) } x , that contains both ξ and ξ (cid:48) . The following proposition gives acharacterization of this quantity in terms of building-walls. We recall that D ( M ) designatesthe dial of building bounded by M and containing x . Proposition 7.18.
Let ξ, ξ (cid:48) be two distinct points in ∂ Γ . Then { ξ | ξ (cid:48) } x = { M ∈ M (Σ) : there exists α (cid:54) = 0 s.t. { ξ, ξ (cid:48) } ⊂ ∂D α ( M ) } . Proof.
Let M , . . . , M k be the set of building-walls such that there exists α (cid:54) = 0 with { ξ, ξ (cid:48) } ⊂ ∂D α ( M ) . Let (cid:96) = { ξ | ξ (cid:48) } x . We prove that k = (cid:96) .For i = 1 , . . . k , let D i be a dial of building bounding M i such that { ξ, ξ (cid:48) } ⊂ ∂D i and x (cid:54)⊂ D i . We set C = D ∩ · · · ∩ D k . Since the building-walls are distinct and ∂C (cid:54) = ∅ itfollows from Fact 7.8 that C is a cone. Let x = proj C ( x ) . As { ξ, ξ (cid:48) } ⊂ ∂C , there existsan infinite minimal gallery starting from x going asymptotically to ξ (resp. ξ (cid:48) ) passingthrough x . Finally, we obtain (cid:96) ≥ d c ( x , x ) ≥ k .Now consider x ∼ x ∼ · · · ∼ x i ∼ · · · (resp. x ∼ x (cid:48) ∼ · · · ∼ x (cid:48) i ∼ · · · ) a minimalinfinite gallery representing ξ (resp. ξ (cid:48) ) in ∂ Γ . Assume that x i = x (cid:48) i for i ≤ (cid:96) and x (cid:96) +1 (cid:54) = x (cid:48) (cid:96) +1 . i = 1 , . . . , (cid:96) let D (cid:48) i be the dial of building such that x i − (cid:54)⊂ D (cid:48) i and x i ⊂ D (cid:48) i . Byminimality of the galleries, we get that { ξ, ξ (cid:48) } ⊂ ∂D (cid:48) i for any index i . Therefore (cid:96) ≤ k andthe proof is finished.In the following, we prove that {·|·} x coincides with a Gromov product in ∂ Γ and thuscontrols a visual metric on ∂ Γ . Proposition 7.19.
Let ξ, ξ (cid:48) be two distinct points in ∂ Γ . Then there exists a bi-infiniteminimal gallery between ξ and ξ (cid:48) that lies at a distance smaller than { ξ | ξ (cid:48) } x + 1 of x .Proof. Let (cid:96) = { ξ | ξ (cid:48) } x and assume that ξ = [ x ∼ x ∼ · · · ∼ x i ∼ · · · ] and ξ (cid:48) = [ x ∼ x (cid:48) ∼ · · · ∼ x (cid:48) i ∼ · · · ] with x i = x (cid:48) i for i ≤ (cid:96) and x (cid:96) +1 (cid:54) = x (cid:48) (cid:96) +1 . We consider two cases. Either x (cid:96) +1 is adjacent to x (cid:48) (cid:96) +1 , or x (cid:96) +1 is not adjacent to x (cid:48) (cid:96) +1 . Inthe first case, Proposition 5.23 implies that the bi-infinite gallery · · · ∼ x (cid:96) +2 ∼ x (cid:96) +1 ∼ x (cid:48) (cid:96) +1 ∼ x (cid:48) (cid:96) +2 ∼ · · · only crosses once the building-walls that separate ξ and ξ (cid:48) . Hence it is minimal.In the second case, we apply the same reasoning to the bi-infinite gallery · · · ∼ x (cid:96) +2 ∼ x (cid:96) +1 ∼ x (cid:96) ∼ x (cid:48) (cid:96) +1 ∼ x (cid:48) (cid:96) +2 ∼ · · · . Finally { ξ | ξ (cid:48) } x or { ξ | ξ (cid:48) } x + 1 is the distance between x and a bi-infinite minimal gallerybetween ξ and ξ (cid:48) . Notation.
Let d ( · , · ) be the self-similar metric on ∂ Γ coming from the geometric action of Γ on DG (Σ) (see Definition 3.2). As a consequence of Proposition 7.19, a general result about hyperbolic spaces due to M.Gromov states that there exist two constants A ≥ and α > such that for any ξ, ξ (cid:48) ∈ ∂ Γ : A − e − α { ξ | ξ (cid:48) } x ≤ d ( ξ, ξ (cid:48) ) ≤ A e − α { ξ | ξ (cid:48) } x . In the sequel we also write d ( ξ, ξ (cid:48) ) (cid:16) e − α { ξ | ξ (cid:48) } x . This means that, d ( ξ, ξ (cid:48) ) is, up to a multiplicative constant, equal to e − α { ξ | ξ (cid:48) } x . An appli-cation of this description of the visual metric on ∂ Γ is the following proposition. Proposition 7.20.
For every (cid:15) > , there exists only a finite set of parabolic limit sets ofdiameter larger than (cid:15) . roof. Let ∂P be a parabolic limit set. Let g (cid:48) Σ I be a residue in Σ such that ∂P (cid:39) ∂ ( g (cid:48) Σ I ) .According to Proposition 7.1, there exists a unique chamber x ⊂ g (cid:48) Σ I such that for everychamber y ⊂ g (cid:48) Σ I there exists a minimal gallery from x to y passing through x . Let g ∈ Γ such that x = gx . Then the diameter of ∂P is controlled by e − α | g | with | g | = d c ( x , gx ) .As there exists only a finite number of g ∈ Γ such that | g | is smaller than a fixed constant,the proposition is proved. ∂ Γ with shadows The following proposition says that shadows are almost balls. This will allow us to constructapproximations using shadows.
Proposition 7.21.
There exists λ > such that for any x ∈ Ch(Σ) with d c ( x , x ) = k there exists z ∈ v x with B ( z, λ − e − αk ) ⊂ v x ⊂ B ( z, λe − αk ) . Proof.
To prove the right-hand side inclusion it is enough to notice that diam v x ≤ Ae − αk where A and α are the visual constants. Let C x be the cone based at x . Let C = D ( M ) ∩· · · ∩ D ( M k ) and g ∈ Γ such that g ( C ) = C x (see Proposition 7.10). Now we recall that g − is a bi-Lipschitz homeomorphism. Restricted to v x , it rescales the metric by a factor e αk . According to Proposition 7.12, there exist r > and z ∈ ∂C such that B ( z, r ) ⊂ ∂C .As there is only a finite number of possible C , the proof is achieved.Let x ∈ Ch(Σ) and v x be the associated shadow as in Definition 7.13. Thanks toProposition 7.19, if d c ( x , x ) = k then diam v x (cid:16) e − αk . We use this property to constructan approximation of ∂ Γ consisting of shadows.For an integer k ≥ we set S k = { x ∈ Ch(Σ) : d c ( x , x ) = k } . The set { v x : x ∈ S k } is a finite covering of ∂ Γ . Now let S (cid:48) k be a subset of S k such that { v x : x ∈ S (cid:48) k } defines a minimal covering of ∂ Γ . This means that for every x ∈ S (cid:48) k thereexists z ∈ v x such that z / ∈ v y for any y ∈ S (cid:48) k \{ x } . Finally we set G k = { v x : x ∈ S (cid:48) k } . In the following, we prove that the sequence { G k } k ≥ defines an approximation of ∂ Γ . Proposition 7.22.
For k ≥ , let S (cid:48) k be the set of chambers previously defined and G k bethe minimal covering of ∂ Γ associated with S (cid:48) k . There exists κ > such that for any x ∈ S (cid:48) k ,there exists ξ x ∈ v x such that: ∀ x ∈ S (cid:48) k : B ( ξ x , κ − e − αk ) ⊂ v x ⊂ B ( ξ x , κe − αk ) , • ∀ x, y ∈ S (cid:48) k with x (cid:54) = y : B ( ξ x , κ − e − αk ) ∩ B ( ξ y , κ − e − αk ) = ∅ . This property is enough to construct an approximation of ∂ Γ . Indeed the visual constant α can be chosen such that / ≤ e − α < . In this case we can extract from { G k } k ≥ asubsequence that is an approximation of ∂ Γ as defined in Subsection 2.1. Proof of Proposition 7.22.
Let x ∈ S (cid:48) k , and let ξ x ∈ v x . With diam v x (cid:16) e − αk , there exists κ > such that for all x ∈ S (cid:48) k : v x ⊂ B ( ξ x , κe − αk ) .We recall that the hyperbolicity provides a constant N ≥ , depending only on thehyperbolicity parameter, such that for x, x (cid:48) ∈ Ch(Σ) with d c ( x , x ) = d c ( x , x (cid:48) ) if d c ( x, x (cid:48) ) ≥ N then v x ∩ v x (cid:48) = ∅ .For any x ∈ S (cid:48) k , we pick z x ∈ v x such that z x / ∈ v y for any y ∈ S (cid:48) k \{ x } . Let x, y ∈ S (cid:48) k , x (cid:54) = y and let c ∈ Ch(Σ) be such that d c ( x , c ) = { z x | z y } x and { z x , z y } ⊂ v c . In this settingwe can write that z x and z y are represented by infinite minimal galleries of the form: • z x = [ x ∼ x ∼ · · · ∼ x i ∼ · · · ∼ x k ∼ x k +1 ∼ · · · ] • z y = [ x ∼ y ∼ · · · ∼ y i ∼ · · · ∼ y k ∼ y k +1 ∼ · · · ] with • x i = c and y i = c for one i ∈ { , . . . , k − } , • x k = x and y k = y .Now we consider two cases. First, we assume that x i +1 is adjacent to y i +1 . In this case,Proposition 5.23 implies that the bi-infinite gallery · · · ∼ x k +1 ∼ x k ∼ · · · ∼ x i +1 ∼ y i +1 ∼ · · · ∼ y k ∼ y k +1 ∼ · · · , only crosses once the building-walls that separate ξ and ξ (cid:48) . Hence it is minimal.Then we assume that x i +1 is not adjacent to y i +1 . In this case we apply the samereasoning to the bi-infinite gallery · · · ∼ x k +1 ∼ x k ∼ · · · ∼ x i +1 ∼ c ∼ y i +1 ∼ · · · ∼ y k ∼ y k +1 ∼ · · · . Summarizing we obtain that one of the following galleries: • · · · ∼ x k +1 ∼ x k ∼ · · · ∼ x i +1 ∼ y i +1 ∼ · · · ∼ y k ∼ y k +1 ∼ · · ·• · · · ∼ x k +1 ∼ x k ∼ · · · ∼ x i +1 ∼ c ∼ y i +1 ∼ · · · ∼ y k ∼ y k +1 ∼ · · ·
55s a bi-infinite minimal gallery from z x to z y .In particular d c ( x k + N , y k + N ) > N and the corresponding shadows do not intersect: v x k + N ∩ v y k + N = ∅ . Now according to Proposition 7.21 there exist ξ x ∈ v x k + N and ξ y ∈ v y k + N such that B ( ξ x , λ − e − α ( k + N ) ) ⊂ v x k + N and B ( ξ y , λ − e − α ( k + N ) ) ⊂ v y k + N . With v x k + N ∩ v y k + N = ∅ , v x k + N ⊂ v x , and v y k + N ⊂ v y we obtain the desired property. The boundary of an apartment is, in a well chosen case, easier to understand than theboundary of the building. This is why we want to compare the modulus in the boundaryof the building with some modulus in the boundary of an apartment.In this section, we start by defining a convenient approximations on ∂ Γ and on theboundaries of the apartments using shadows and retraction maps. Afterwards, we introducethe weighted modulus on the boundary of an apartment. Then we prove Theorem 8.9.This theorem is, after Theorem 6.12, the second major step in proving the main theorem(Theorem 10.1). It states that weighted modulus are comparable to the modulus in ∂ Γ .Finally, using the ideas in Subsection 3.2, we reveal a connection between the conformaldimension of ∂ Γ and a critical exponent computed in the boundary of an apartment.We use the notation and assumptions from Sections 5, 6 and 7. In particular p ≥ isa fixed constant. We fix Γ the graph product associated with the pair ( G , { Z /q i Z } i =1 ,...,n ) .The self-similar metric d ( · , · ) on ∂ Γ is defined as in Subsection 7.3. The visual exponent of d ( · , · ) is α . As in Section 3, d denotes a small constant compared with diam ∂ Γ and withthe constant of approximate self-similarity. Then F is the set of curves of diameter largerthan d . ∂A and in ∂ Γ In the rest of this article we fix an apartment A containing the base chamber x . We shallconnect the geometry and the modulus in ∂A and in ∂ Γ . Naturally we will use in ∂A andin ∂ Γ the same concepts. Here we summarize some of the notation used in the following toavoid confusion. First we write A p (Σ) = { B ∈ A p (Σ) : x ⊂ B } . π denote the retraction π A,x : Σ −→ A . We also denote by π the extension of theretraction to the boundary. The notation d ( · , · ) and α are also used to describe the metricon ∂B for any B ∈ A p (Σ) .An apartment is a thin building, so we can use in ∂A the tools presented in Subsections7.2 and 7.3. First, we define on ∂A a combinatorial self-similar metric as in Subsection 7.3.Since x ⊂ A , for ξ, ξ (cid:48) ∈ ∂A , the quantity { ξ | ξ (cid:48) } x is the same whether we compute it in A or in Σ . Hence, if we choose the same visual exponents for the visual metric in ∂ Γ and thevisual metric in ∂A , then the metrics coincide up to a multiplicative constant. In the rest, d ( · , · ) designate the metric on both ∂A and ∂ Γ . Likewise, α and A designate the visualconstants or both ∂A and ∂ Γ .Finally, it makes sense to talk about cones of chambers in A and shadows in ∂A . Theresults of 7.2 also hold in ∂A . Notation. • for ξ ∈ ∂A and r > we designate by B ( ξ, r ) ⊂ ∂ Γ the open ball of ∂ Γ of radius r and center ξ , • for x ∈ Ch( A ) we write C x for the cone of chambers based on x in Σ , • for x ∈ Ch( A ) we write v x (resp. w x ) for the shadow of x in ∂ Γ (resp. ∂A ). Usually we will use the following conventions. • v (resp. w ) designates an open subset of ∂ Γ (resp. of ∂A ), • ∂P (resp. ∂Q ) designates a parabolic limit set in ∂ Γ (resp. in ∂A ). The following lemma says that shadows have a nice behavior under retraction maps.
Lemma 8.1.
Let A ∈ A p (Σ) and let x ∈ Ch(Σ) and v x be the associated shadow in ∂ Γ asdefined in Definition 7.13. Then • either x / ∈ Ch( A ) and Int( v x ) ∩ ∂A = ∅ , • or x ∈ Ch( A ) and v x ∩ ∂A is a shadow in ∂A .In the second case v x ∩ ∂A = π ( v x ) .Proof. Let C x be the cone based on x . If Int( v x ) ∩ ∂A (cid:54) = ∅ then there exists a chamber c in A ∩ C x . By convexity, a minimal gallery from x to c that passes through x is included in A and x ⊂ A . Therefore v x ∩ ∂A is the shadow in ∂A associated with x .57e fix { G Ak } k ≥ an approximation of ∂A based on shadows as constructed in Subsection7.4. Notation.
For k ≥ we set G k := { v y ⊂ ∂ Γ : π ( v y ) ∈ G Ak } . We recall that we chose the same visual exponents for the metrics in ∂ Γ and ∂A . As aconsequence of Lemma 8.1 we get the following fact. Fact 8.2.
There exists κ > such that { G Ak } k ≥ and { G k } k ≥ are κ -approximations.Moreover, for any w ∈ G Ak there exists a unique (cid:101) w ∈ G k such that Int( (cid:101) w ) ∩ ∂A (cid:54) = ∅ and π ( (cid:101) w ) = w . Hereafter, { G k } k ≥ designates the approximation of ∂ Γ obtained from { G Ak } k ≥ thanksto the preceding fact. This approximation of ∂ Γ is canonically associated with { G Ak } k ≥ inthe following sense: from { G k } k ≥ we can equip any B ∈ A p (Σ) with an approximationisometric to { G Ak } k ≥ . Indeed if B ∈ A p (Σ) , for k ≥ we set G Bk := { w = ∂B ∩ v : v ∈ G k } . Now let B ∈ A p (Σ) and f : B −→ A be the type preserving isometry that fixes x . Themap f is realized by the restriction to B of the retraction π and we get the following fact. Fact 8.3. G Ak = { f ( v ) } v ∈ G Bk . Now that an approximation { G k } k ≥ is fixed the results we will obtain on the combi-natorial modulus in ∂ Γ will be valid, up to multiplicative constants, for any approximationthanks to Proposition 2.5. ∂A On scale k ≥ , to compare the modulus in the building with the modulus in the apartmentwe need to compare the cardinality of G k with the cardinality of G Ak . If the building isthick these quantities differ by an exponential factor in k . This is the reason we attach aweight to the elements of G Ak . Definition 8.4.
Let w ∈ G Ak , we set q ( w ) = { v ∈ G k : π ( v ) = w } . Let k ≥ and let F A be a set of curves contained in ∂A . As in Subsection 2.1, a positivefunction ρ : G Ak −→ [0 , + ∞ ) is said to be F A -admissible if for any γ ∈ F A (cid:88) γ ∩ w (cid:54) = ∅ ρ ( w ) ≥ . weighted p -mass of ρ in ∂A is W M Ap ( ρ ) = (cid:88) w ∈ G Ak q ( w ) ρ ( w ) p . Definition 8.5.
Let k ≥ and let F A be a set of curves contained in ∂A , we define the weighted G Ak -combinatorial p -modulus of F A by Mod Ap ( F A , G Ak ) := inf { W M Ap ( ρ ) } , where the infimum is taken over the set of F A -admissible functions and with the convention Mod Ap ( ∅ , G Ak ) = 0 . For simplicity, we usually use the terminology weighted modulus . We can check that Proposition 2.3 holds for weighted modulus as well and the proof isidentical to the one for the usual combinatorial modulus.This definition of the weighted modulus strongly depends on the choice we have madefor the approximation. In particular, it does not permit to compute a weighted modulusrelatively to a generic approximation of ∂A . As a consequence, an analogue to Proposition2.5 would make no sense here. This is a huge restriction on the use of the weighted modulus.Indeed, this proposition is essential in proving Proposition 3.11 and Theorem 6.12 for theusual combinatorial modulus.However, in the rest of the paper, the weighted modulus will be used to compute in-equalities for the usual combinatorial modulus. As the usual combinatorial modulus, upto multiplicative constant, does not depend on the choice of the approximation, this pointwill not be a problem for us.The following proposition says that the weights are given by the types of the building-walls crossed by a minimal gallery. We recall that the group Γ is given by the followingpresentation Γ = (cid:104) s i ∈ S | s q i i = 1 , s i s j = s j s i if v i ∼ v j (cid:105) . Proposition 8.6.
Let w ∈ G Ak be such that w is a shadow w = w x for x ∈ Ch( A ) . Let x ∼ s x ∼ s · · · ∼ s k − x k − ∼ s k x be a minimal gallery where s i is the generator of Γ associated with the type of the building-wall between x i − and x i for every ≤ i ≤ k . If q i is the order of s i for every ≤ i ≤ k then q ( w ) = (cid:89) i =1 ,...,k q i − . Proof.
Let w ∈ G Ak and x ∈ Ch( A ) be such that w = w x in ∂A . Then we observethat { v ⊂ ∂ Γ : π ( v ) = w } = { v y ⊂ ∂ Γ : π ( y ) = x } . As a consequence, we obtain q ( w ) = π − ( x ) .Now consider the gallery x ∼ s x ∼ s · · · ∼ s k − x k − ∼ s k x given in the statement ofproposition. Since π preserves the types, y ∈ Ch(Σ) is in π − ( x ) if and only if there exists a59inimal gallery from x to y in Σ of the form x ∼ s y ∼ s · · · ∼ s k − y k − ∼ s k y . Finally,we obtain q ( w ) = (cid:81) i =1 ,...,k q i − .Thanks to the choices we have made, the weighted modulus is invariant up to a changeof apartment in the following sense. For B ∈ A p (Σ) consider the approximation G Bk given by Fact 8.3. To any element w ∈ G Bk we attach a weight and define a weighted G Bk -combinatorial p -modulus as it is done in ∂A . Now let f : B −→ A be a type preservingisometry that fixes x and denote f : ∂B −→ ∂A the extension of this map to the boundary.The map f is realized by the restriction of the retraction π to B . Thus f preserves theweights. Then the following fact is an immediate consequence of Fact 8.3. Fact 8.7.
Let B ∈ A p (Σ) . Then for any k ≥ and any set of curves F B contained in ∂B one has Mod Bp ( F B , G Bk ) = Mod Ap ( f ( F B ) , G Ak ) . Note that, for any k ≥ and any w ∈ G Ak one has ≤ q ( w ) ≤ ( q − k with q := max { q , . . . , q n } . Therefore for any set of curves F A contained in ∂A , the next inequalities follow directlyfrom the definition mod Ap ( F A , G Ak ) ≤ Mod Ap ( F A , G Ak ) ≤ ( q − k mod Ap ( F A , G Ak ) , where the modulus in small letters designates the usual modulus computed in ∂A . Inparticular if Γ is of constant thickness q ≥ then Mod Ap ( F A , G Ak ) = ( q − k mod Ap ( F A , G Ak ) . As a consequence, at fixed scale k ≥ , the weighted modulus depends only on the boundaryof an apartment. We will discuss this particular case in Sections 9 and 10.The following proposition is a major motivation of the definition of the weighted mod-ulus. Proposition 8.8.
Let F be a set of curves in ∂ Γ and let F A be a set of curves in ∂A suchthat π ( F ) ⊂ F A . Then Mod p ( F , G k ) ≤ Mod Ap ( F A , G Ak ) . Proof.
Let ρ A be a F A -admissible function. We set ρ : G k −→ [0 , + ∞ ) defined by ρ ( v ) = ρ A ◦ π ( v ) . If γ ∈ F , let γ A := π ◦ γ . Then, as γ A ∈ F A L ρ ( γ ) = (cid:88) v ∩ γ (cid:54) = ∅ ρ A ◦ π ( v ) ≥ (cid:88) w ∩ γ A (cid:54) = ∅ ρ A ( w ) ≥ , ρ is F -admissible. Furthermore, one has: M p ( ρ ) = (cid:88) v ∈ G k ρ A ◦ π ( v ) p = (cid:88) w ∈ G Ak q ( w ) · ρ A ( w ) p = W M Ap ( ρ A ) . With the first point it follows that
Mod p ( F , G k ) ≤ Mod Ap ( F A , G Ak ) . ∂ Γ compared with weighted modulus in ∂A As before d > is a small constant compared with diam ∂ Γ and with the constant ofapproximate self-similarity.We recall that the apartment A ∈ A p (Σ) is fixed. Thanks to Fact 8.7 the followingresult hold for any apartment containing x .In this subsection we continue to use the approximations G k and G Ak defined at thebeginning of Subsection 8.2. We recall that if η is a non-constant curve of ∂ Γ , the notation U (cid:15) ( η ) designates the (cid:15) -neighborhood of η relative to the C topology. If η is a non-constantcurve contained in ∂A , we use the notation U A(cid:15) ( η ) := { γ ∈ U (cid:15) ( η ) : γ ⊂ ∂A } . The next theorem says that in this case, the modulus of U (cid:15) ( η ) in the boundary of thebuilding is controlled by the weighted modulus of U A(cid:15) ( η ) in the boundary of the apartment.It is a key step in the proof of Theorem 10.1. Theorem 8.9.
Let p ≥ , let η ∈ F and assume η ⊂ ∂A . For (cid:15) > small enough so thatthe hypothesis of Theorem 6.12 hold in ∂ Γ , there exists a positive constant C = C ( p, η, (cid:15) ) independent of k such that for every k ≥ large enough Mod p ( U (cid:15) ( η ) , G k ) ≤ Mod Ap ( U A(cid:15) ( η ) , G Ak ) ≤ C · Mod p ( U (cid:15) ( η ) , G k ) . Furthermore, when p belongs to a compact subset of [1 , + ∞ ) the constant C may be chosenindependent of p . For the rest of the subsection η ∈ F and (cid:15) > are as in the hypothesis of the precedingtheorem. For each η ∈ F we fix a constant r > such that the hypothesis of Theorem6.12 are satisfied. To prove the theorem we need to introduce the following notation: • Aut Σ is the full group of type preserving isometries of Σ . • For n ≥ , B n ⊂ Ch(Σ) is the ball of center x and of radius n for the distance overthe chambers d c ( · , · ) . • For n ≥ , K n < Aut Σ is the pointwise stabilizer of B n under the action of Aut Σ . • F n := { gγ ⊂ ∂ Γ : g ∈ K n and γ ∈ U A(cid:15) ( η ) } .61he main step to prove the theorem is to show that F n is an intermediate set of curvesbetween U A(cid:15) ( η ) and U (cid:15) ( η ) . This will be done in Lemma 8.11. Before proving this, we needto discuss the action of K n on the chambers. The next lemma uses ideas of [Cap14, Lemma3.5 and Proposition 8.1]. Lemma 8.10.
There exists an integer
N > depending only on n and satisfying thefollowing property. Let x ∈ Ch(Σ) , set d c ( x , x ) = k and assume k > n . Let x ∼ s x ∼ s · · · ∼ s n x n ∼ s n +1 · · · ∼ s k − x k − ∼ s k x be a minimal gallery where s , . . . , s k is the family of types of the building-walls crossed bythis gallery. Then q − N · k (cid:89) i = n +1 q i − ≤ K n .x ≤ k (cid:89) i = n +1 q i − , where q := max { q , . . . , q n } .Proof. Since K n preserves the types and fixes x , . . . , x n it follows that K n .x ≤ k (cid:89) i = n +1 q i − . Now for D ∈ D (Σ) we write U ( D ) the pointwise stabilizer of D under the action of Aut Σ and we set U ( n ) = (cid:104) U ( D ) | B n ⊂ Ch( D ) (cid:105) . Clearly U ( n ) < K n and K n .x ≥ U ( n ) .x. Now if we write M i the building-wall between x i and x i +1 , we observe that the orbit of x i +1 under U ( D ( M i )) has q i − elements. Indeed, U ( D ( M i )) acts as the full group ofpermutations on the set { D ( M i ) , . . . , D q i − ( M i ) } .Note that U ( D ( M i )) < U ( n ) if and only if B n ⊂ Ch( D ( M i )) . Otherwise M i crosses B n , because x ∈ Ch( D ( M i )) . As a consequence, we set N the number of building-wallsthat cross B n and we obtain U ( n ) .x ≥ q − N · k (cid:89) i = n +1 q i − . This achieves the proof.Now we can prove the main lemma. 62 emma 8.11.
Let p ≥ . For n ≥ large enough, there exist two positive constants C , C depending only on p , η , (cid:15) , such that for every k > n : Mod Ap ( U A(cid:15) ( η ) , G Ak ) ≤ C · Mod p ( F n , G k ) ≤ C · Mod p ( U (cid:15) ( η ) , G k ) . Furthermore, when p belongs to a compact subset of [1 , + ∞ ) the constants may be chosenindependent of p .Proof. i) First we prove the right-hand side inequality. According to Proposition 7.19,for any g ∈ K n and any ξ ∈ ∂ Γ , d ( ξ, gξ ) ≤ A.e − αn where A is the visual multiplicativeconstant. Hence for n ≥ large enough, by triangular inequality, F n ⊂ U (cid:15) ( η ) . Then, asa consequence of Theorem 6.12, for a fixed r > the combinatorial modulus of U (cid:15) ( η ) iscontrolled by the combinatorial modulus of U (cid:15) ( η ) with multiplicative constants dependingonly on p , η , (cid:15) . Thus, by Proposition 2.3 (1), there exists C = C ( p, η, (cid:15) ) such that Mod p ( F n , G k ) ≤ C · Mod p ( U (cid:15) ( η ) , G k ) . ii) Now we fix an integer n ≥ large enough so that the first part of the proof holds.We use the notation K := K n for simplicity. Moreover we assume that k > n . Let ρ : G k −→ [0 , + ∞ ) be a minimal F n -admissible function and set ρ A : G Ak −→ [0 , + ∞ ) thefunction defined by: ρ A ( w ) = (cid:90) K ρ ( g (cid:101) w ) dµ ( g ) , where µ denotes the Haar probability measure over K and where the function : w ∈ G Ak −→ (cid:101) w ∈ G k is given by Fact 8.2. Let w ∈ G Ak and let x ∈ Ch(Σ) be such that v x = (cid:101) w . Then d c ( x , x ) = k . As in Proposition 8.6, let x ∼ s x ∼ s · · · ∼ s n x n ∼ s n +1 · · · ∼ s k − x k − ∼ s k x be a minimal gallery where s , . . . , s k is the family of types of the building-walls crossed bythis gallery. We set q ( w, n ) = (cid:89) i = n +1 ,...,k q i − . We notice that for any g ∈ K the translated g (cid:101) w = gv x is the shadow v gx . In particular,this means that K. (cid:101) w = K.x . Then according to Lemma 8.10 ( ∗ ) q ( w, n )( q − N ≤ K. (cid:101) w ≤ q ( w, n ) , where q := max { q , . . . , q n } and N is the number of building-walls crossing B n .As a consequence we can write ρ A ( w ) = 1 K. (cid:101) w · (cid:88) v ∈ K. (cid:101) w ρ ( v ) , γ ∈ U A(cid:15) ( η ) : L ρ A ( γ ) = (cid:88) w ∩ γ (cid:54) = ∅ (cid:90) K ρ ( g (cid:101) w ) dµ ( g ) = (cid:90) K (cid:88) w ∩ γ (cid:54) = ∅ ρ ( g (cid:101) w ) dµ ( g ) = (cid:90) K (cid:88) v ∩ g ( γ ) (cid:54) = ∅ ρ ( v ) dµ ( g ) . Since g ( γ ) ∈ F n , we get (cid:80) v ∩ g ( γ ) (cid:54) = ∅ ρ ( v ) ≥ and ρ A is F A -admissible.On the other hand, thanks to Jensen’s inequality, for p ≥ one has: W M Ap ( ρ A ) ≤ (cid:88) w ∈ G Ak q ( w ) (cid:90) K ρ ( g (cid:101) w ) p dµ ( g ) = (cid:88) w ∈ G Ak q ( w ) K. (cid:101) w · (cid:88) v ∈ K. (cid:101) w ρ ( v ) p . Hence with ( ∗ ) we obtain W M Ap ( ρ A ) ≤ (cid:88) w ∈ G Ak ( q − N · q ( w ) q ( w, n ) · (cid:88) v ∈ K. (cid:101) w ρ ( v ) p ≤ ( q − n + N M p ( ρ ) . Finally we get:
Mod Ap ( U A(cid:15) ( η ) , G Ak ) ≤ ( q − n + N Mod p ( F n , G k ) . This last multiplicative constant depends only on n and on the geometry of the building.Since n depends only on η and (cid:15) the second inequality is proved.iii) The last statement of the lemma is an immediate consequence of the two first partsand of Theorem 6.12. Proof of Theorem 8.9.
Since π ( U (cid:15) ( η )) ⊂ U A(cid:15) ( η ) , Proposition 8.8 and Lemma 8.11 imply thetheorem. Here we continue to use the approximations G k and G Ak defined in Subsection 8.2. For η a non-constant curve in ∂A , ∂Q a parabolic limit set in ∂A , and δ, r, (cid:15) > , we use thefollowing notation: • F A = { γ ∈ F : γ ⊂ ∂A } , • F Aδ,r ( ∂Q ) is the subset of F A consisting of all curves γ satisfying: – γ ⊂ N δ ( ∂Q ) , – γ (cid:54)⊂ N r ( ∂Q (cid:48) ) for any connected parabolic limit set ∂Q (cid:48) (cid:32) ∂Q ,64 δ ( · ) refers to the increasing function in Theorem 6.12.We recall that the apartment A ∈ A p (Σ) is fixed. Thanks to Fact 8.7, the followingresult holds for any apartment containing x .The main consequence of the previous subsection is Theorem 8.13, where we controlfrom above the combinatorial modulus of F by the weighted-combinatorial modulus of F A . Lemma 8.12.
Let p ≥ and A ∈ A p (Σ) . Let ∂P be a parabolic limit set in ∂ Γ andassume that x ⊂ Conv( ∂P ) . Let γ be a non-constant curve in ∂Q = ∂P ∩ ∂A such that ∂Q is the smallest parabolic limit set of ∂A containing γ . Let r > be small enough sothat γ (cid:54)⊂ N r ( ∂Q (cid:48) ) for any connected parabolic limit set ∂Q (cid:48) (cid:32) ∂Q . Let δ < δ ( r ) . Then for (cid:15) > small enough, there exists a constant C = C ( p, γ, r, (cid:15) ) such that for every k ≥ p ( F δ,r ( ∂P ) , G k ) ≤ C · Mod p ( U (cid:15) ( γ ) , G k ) ≤ C · Mod Ap ( F Aδ,r ( ∂Q ) , G Ak ) . In particular
Mod p ( F δ,r ( ∂P ) , G k ) ≤ C · Mod Ap ( F A , G Ak ) . Furthermore, when p belongs to a compact subset of [1 , + ∞ ) the constant C may be chosenindependent of p .Proof. As in the beginning of the proof of Theorem 6.12 we can assume, without loss ofgenerality, that for (cid:15) > small enough • x ⊂ Conv( γ ) for every γ ∈ F , • U (cid:15) ( γ ) ⊂ F .As before, the multiplicative constants resulting from these assumptions only depends on d . Now for (cid:15) > small enough, we obtain by Proposition 2.3(1) and the preceding assump-tion Mod Ap ( U A(cid:15) ( γ ) , G Ak ) ≤ Mod Ap ( F Aδ,r ( ∂Q ) , G Ak ) ≤ Mod Ap ( F A , G Ak ) . Since π ( U (cid:15) ( γ )) ⊂ U A(cid:15) ( γ ) , with Proposition 8.8 one has Mod p ( U (cid:15) ( γ ) , G k ) ≤ Mod Ap ( U A(cid:15) ( γ ) , G Ak ) . Finally thanks to Theorem 6.12 there exists C = C ( p, γ, r, (cid:15) ) such that for every k ≥ p ( F δ,r ( ∂P ) , G k ) ≤ C · Mod p ( U (cid:15) ( γ ) , G k ) . Theorem 8.13.
For any p ≥ , there exists a constant D = D ( p ) such that for every k ≥ p ( F , G k ) ≤ D · Mod Ap ( F A , G Ak ) . Proof. i) First, we recall the following notation. For δ, r > and for ∂P a connectedparabolic limit set, F δ,r ( ∂P ) is the set of all the curves in F satisfying: • γ ⊂ N δ ( ∂P ) , • γ (cid:54)⊂ N r ( ∂P (cid:48) ) for any connected parabolic limit set ∂P (cid:48) (cid:32) ∂P .Now, as it is done in [BK13, as a remark of Corollary 6.2.] in boundaries of Coxetergroups, we observe that F splits into a finite union F = F δ ,r ( ∂P ) ∪ · · · ∪ F δ N ,r N ( ∂P N ) with δ i < δ ( r i ) .Indeed, let P = { ∂P , . . . , ∂P N } be the finite set (see Proposition 7.20) of all theparabolic limit sets of diameter larger than d . For ∂P ∈ P we call height of ∂P themaximal length of a sequence in P of the form ∂P (cid:48) (cid:32) ∂P (cid:48) (cid:32) · · · (cid:32) ∂P (cid:48) i = ∂P. Now, we index P thanks to the height P = { ∂P , , . . . , ∂P ,N , . . . , ∂P i,j , . . . , ∂P M, , . . . , ∂P M,N M } where i is the height of ∂P i,j . We fix a small r > and δ < δ ( r ) . Then by induction onthe height we set r i +1 = δ i and δ i +1 < min { δ ( r i +1 ) , δ i } . Now let ∂P ∈ P be of height i > . By construction, for any i (cid:48) < i we have r i < δ i (cid:48) .Hence for any ∂P (cid:48) (cid:32) ∂P of height i (cid:48) we have N r i ( ∂P (cid:48) ) ⊂ N δ i (cid:48) ( ∂P (cid:48) ) and F = M (cid:91) i =0 N i (cid:91) j =1 F δ i ,r i ( ∂P i,j ) . ii) Let ∂P be one of the parabolic limit sets appearing in the preceding decomposition of F and δ, r > be the corresponding constants. As in the beginning of the proof of Theorem6.12, we can assume that x ⊂ Conv( ∂P ) . Again the multiplicative constant resulting fromthis assumption only depends on d . Now pick B ∈ A p (Σ) such that ∂B ∩ ∂P (cid:54) = ∅ and fix66 curve γ and (cid:15) > so that the hypothesis of Lemma 8.12 are satisfied. Then there exists C = C ( p, γ, r, (cid:15) ) such that for every k ≥ p ( F δ,r ( ∂P ) , G k ) ≤ C · Mod Bp ( F B , G B ) . iii) With Fact 8.7, we observe that the weighted modulus on the right-hand side of thepreceding inequality is independent of the choice of B ∈ A p (Σ) . Finally, by Proposition2.3 (2) and the two first parts of this proof, there exists a constant D = D ( p ) such thatsuch that for every k ≥ p ( F , G k ) ≤ D · Mod Ap ( F A , G Ak ) . Note that for the moment we cannot prove a converse inequality between the modulus.Indeed, in the proof of Lemma 8.12 the use of Theorem 6.12 is a key point. As we saidbefore, we cannot prove an analogue of Theorem 6.12 for the weighted modulus.Nevertheless, we can define a critical exponent in connection with the weighted modulusas it is done in Subsection 3.2. Then Theorem 8.13 can be used to helps us understand thisnew critical exponent.
Proposition 8.14.
There exists p ≥ such that for p ≥ p the weighted modulus Mod Ap ( F A , G Ak ) goes to zero as k goes to infinity.Proof. This proof is the same as the proof of Proposition 3.4. We recall that κ is theconstant of the approximations { G k } k ≥ and { G Ak } k ≥ .According to the doubling condition and the definition of an approximation, there existsan integer N (cid:48) such that each element w ∈ G Ak is covered by at most N (cid:48) elements of G Ak +1 .As a consequence, if K > is the cardinality G , then G Ak ≤ K · N (cid:48) k for any k ≥ . Moreover, as we saw in the proof of Proposition 2.4, there exists a constant K (cid:48) > suchthat the constant function ρ : v ∈ G Ak −→ ρ ( v ) = K (cid:48) · − k ∈ [0 , + ∞ ) is F A -admissible.As a consequence mod Ap ( F A , G Ak ) ≤ C · (cid:16) N (cid:48) p (cid:17) k , where C is a positive constant. Hence we obtain Mod Ap ( F A , G Ak ) ≤ ( q − k · mod Ap ( F A , G Ak ) ≤ C · (cid:16) ( q − N (cid:48) p (cid:17) k , Thus, for p large enough, Mod Ap ( F A , G Ak ) goes to zero.It is now natural to define a critical exponent for the weighted modulus in the apartment.67 efinition 8.15. The critical exponent Q W of the weighted modulus in ∂A is defined asfollows Q W = inf { p ∈ [1 , + ∞ ) : lim k → + ∞ Mod Ap ( F A , G Ak ) = 0 } . To avoid confusion, we use the following notation • Q for the critical exponent associated with the usual modulus Mod p ( · , G k ) in ∂ Γ , • Q A for the critical exponent associated with the usual modulus mod Ap ( · , G Ak ) in ∂A , • Q W for the critical exponent associated with the weighted modulus Mod Ap ( · , G Ak ) in ∂A .We recall that Q and Q A are respectively the conformal dimension of ∂ Γ and of ∂A (cid:39) ∂W (see Theorem 3.7). The inequalities between the different modulus imply the followingcorollary. Corollary 8.16.
The following inequalities hold Q A ≤ Q ≤ Q W . Proof.
With Proposition 2.3 (1) and Theorem 8.13, one has mod Ap ( F A , G Ak ) ≤ Mod p ( F , G k ) ≤ D · Mod Ap ( F A , G Ak ) . The inequalities between the critical exponents follow.
In this section we use the notation and the assumptions from the previous section. Inparticular, the self-similar metric on ∂ Γ is d ( · , · ) . We fix d a small constant comparedwith diam ∂ Γ and with the constant of approximate self-similarity. Then F is the set ofcurves of diameter larger than d . The notation δ ( · ) still refers to the increasing functionin Theorem 6.12.As before we fix an apartment A ∈ A p (Σ) and F A is the set of curves in ∂A of diameterlarger than d .We assume that Σ is of constant thickness q ≥ . This means that Γ is the graphproduct given by the pair ( G , { Z /q Z } i =1 ,...,n ) . As before { G Ak } k ≥ and { G k } k ≥ are theapproximations of ∂A and ∂ Γ provided by Fact 8.2. We already noticed that, with theconstant thickness assumption, we obtain for k ≥ and F A a set of curves contained in ∂A Mod Ap ( F A , G Ak ) = q k mod Ap ( F A , G Ak ) , ∂A . In par-ticular, this means that from Theorem 6.12 applied to mod Ap ( · , G Ak ) we can obtain analogousinequalities for Mod Ap ( · , G Ak ) .The constant thickness allows us to control by bellow the combinatorial modulus of F by the weighted-combinatorial modulus of F A . Combining with the results of Subsection8.5, we obtain a full control of the two modulus. Theorem 9.1.
For any p ≥ , there exists a constant D = D ( p ) such that for every k ≥ D − · Mod Ap ( F A , G Ak ) ≤ Mod p ( F , G k ) ≤ D · Mod Ap ( F A , G Ak ) . In particular Q W = Q .Proof. The right-hand side inequality is given by Theorem 8.13. The proof is almost thesame for the left-hand side inequality. Indeed, F A admits a decomposition analogous tothe decomposition used at the beginning of the proof of Theorem 8.13. With a fixed suchdecomposition and with Proposition 2.3 (2), it is sufficient to prove that for any paraboliclimit set ∂Q ⊂ ∂A and any δ, r > with δ < δ ( r ) , there exists a constant C = C ( p, ∂Q, r ) such that for every k ≥ Ap ( F Aδ,r ( ∂Q ) , G Ak ) ≤ C · Mod p ( F , G k ) . To this end, fix η a non-constant curve in ∂Q and (cid:15) > such that the hypotheses ofTheorem 6.12 in ∂A and of Theorem 8.9 are satisfied. Then there exist two constants K and K (cid:48) depending only on p , η , r and (cid:15) such that for every k ≥ Ap ( F Aδ,r ( ∂Q ) , G Ak ) ≤ K · Mod Ap ( U A(cid:15) ( η ) , G Ak ) ≤ K (cid:48) · Mod p ( U (cid:15) ( η ) , G k ) . Finally, as in the beginning the proof of Theorem 6.12 we can assume without loss ofgenerality that for (cid:15) > small enough U (cid:15) ( η ) ⊂ F . Again the multiplicative constantresulting from this assumption only depends on d . This assumption and Proposition 2.3(1) provide the desired inequality.The equality of the critical exponents is an immediate consequence of the inequalitiesbetween the modulus. Remark 9.2.
In the case where Σ is a right-angled Fuchsian building of constant thickness,M. Bourdon gave the explicit value of the conformal dimension of ∂ Γ . Theorem 9.3 ([Bou97]) . Let Γ be the graph product associated with a pair ( C n , { Z /q Z } i =1 ,...,n ) where C n is a cyclic graph of length n ≥ and q ≥ , then Confdim( ∂ Γ) = 1 + log( q − n − . In a well chosen case, the symmetries of the Davis chamber, that extend to the boundaryof an apartment, provide a strong control of the weighted modulus. This lead to the proofof the main theorem of this article.Here we still assume that Γ is of constant thickness q ≥ . As usual, W is the Coxetergroup, associated with Γ . As before { G Ak } k ≥ and { G k } k ≥ are the approximations of ∂A and ∂ Γ provided by Fact 8.2.In this subsection, we assume that W is the group generated by the reflections aboutthe faces of a compact right-angled polytope D ⊂ H d .Now we write Ref( H d ) the group generated by all the hyperbolic reflections in H d . Inthe following we designate by Ref( D ) the stabilizer of D under the action of Ref( H d ) .Now, with additional assumptions on the regularity of D , we prove that ∂ Γ satisfies theCLP. Theorem 10.1.
Let Γ be a graph product of constant thickness q ≥ . Assume that W is the group generated by the reflections about the faces of a compact right-angled polytope D ⊂ H d . Moreover, assume that the quotient of D by R ef ( D ) is a simplex in H d . Then ∂ Γ satisfies the CLP. Now we assume that the hypotheses of the preceding theorem hold and we use thefollowing notation.
Notation. • T is the hyperbolic simplex in H d isometric to D/R ef ( D ) . • W T is the hyperbolic reflection group generated by the reflections about the codimension1 faces of T . We notice that W is a finite index subgroup of W T . Indeed, W is a subgroup of W T and they both act discretely on H d with finite co-volume. Then W T acts by polyhedralisometries on an apartment of Σ . Indeed, a reflection about a face of T either preserves D ,or is a reflection about a face of D . In particular, it preserves the tilling of H d by D .Thanks to the constant thickness and the results in the preceding section, we only needto study the usual combinatorial modulus in the apartment to prove the theorem. Lemma 10.2.
Let p ≥ and let A ∈ A p (Σ) . Let η be a non-constant curve in ∂A . Thenthere exists a constant C = C ( p, η, (cid:15) ) such that for every k ≥ Ap ( F A , G Ak ) ≤ C · mod Ap ( U A(cid:15) ( η ) , G Ak ) . Furthermore, when p belongs to a compact subset of [1 , + ∞ ) the constant C may be chosenindependent of p . βγ δ θω Figure 8: If D is a dodecahedron, T is the hyperbolic tetrahedron with dihedral angles α = π/ , β = π/ , γ = δ = ω = π/ and θ = π/ , Proof.
To prove this lemma, we first use the fact that ∂W T is identified with ∂A as W T acts geometrically on A . Hence the combinatorial visual metric on ∂A defines a self-similarmetric d W T on ∂W T . A κ -approximation { G Ak } k ≥ of ∂A induces a κ -approximation on ∂W T with same the modulus.On the other hand, ∂W T contains no proper parabolic limit set. Indeed, the Coxeterpolytope T of W T is a simplex so all the proper parabolic subgroups of W T are finite. Inparticular, for any non-constant curve η ⊂ ∂W T , the smallest parabolic subset containing η is ∂W T .As a consequence, by [BK13, Corollary 6.2.] we get that for every (cid:15) > , there exists C = C ( p, η, (cid:15) ) such that for every k ≥ Ap ( F A , G Ak ) ≤ C · mod Ap ( U A(cid:15) ( η ) , G Ak ) . The Corollary 6.2 in [BK13] is the equivalent for Coxeter groups of our Theorem 6.12 forgraph products.
Proof of Theorem 10.1.
We check that the hypotheses of Proposition 3.12 are satisfied. Toprove that
Mod ( F , G k ) is unbounded, it is enough to prove that there exist N disjointcurves of diameter larger than d in ∂ Γ for every N ∈ N as we did at the beginning of theproof of Theorem 6.13.Now we let p ≥ , η be a non-constant curve in ∂ Γ , and (cid:15) > . Without loss of generality,we can assume that there exists A ∈ A p (Σ) such that η ⊂ ∂A . Indeed, as a consequence ofTheorem 6.12 if ∂P is the smallest parabolic limit set containing η . For η (cid:48) a curve such that ∂P is the smallest parabolic limit set containing η (cid:48) , then, up to a multiplicative constant,the behavior of Mod p ( U (cid:15) ( η ) , G k ) and Mod p ( U (cid:15) ( η (cid:48) ) , G k ) when k goes to infinity are the same.The multiplicative constant resulting from this assumption only depends on η .Using the same arguments as in the beginning of the proof of Theorem 6.12, we canalso assume, without loss of generality, that for (cid:15) > small enough • U (cid:15) ( η ) ⊂ F , 71 x ∈ Ch( A ) .Again the multiplicative constant resulting from these assumptions only depends on d .Then, thanks to the constant thickness, the inequality of Lemma 10.2 becomes Mod Ap ( F A , G Ak ) ≤ C · Mod Ap ( U A(cid:15) ( η ) , G Ak ) , where C depends only on p , η and (cid:15) .Finally, it is enough to apply Theorem 8.13 to the left-hand term and Theorem 8.9 tothe right-hand term of the previous inequality to complete the proof. Corollary 10.3.
Let Σ be a building of constant thickness q ≥ . Assume that the Coxetergroup of Σ is the reflection group of the right-angled dodecahedron in H or the reflectiongroup of the right-angled 120-cells in H , then ∂ Σ satisfies the CLP. Remark 10.4.
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