Helly groups, coarse Helly groups, and relative hyperbolicity
aa r X i v : . [ m a t h . G R ] J a n HELLY GROUPS, COARSE HELLY GROUPS, AND RELATIVEHYPERBOLICITY
DAMIAN OSAJDA AND MOTIEJUS VALIUNAS
Abstract.
A simplicial graph is said to be ( coarse ) Helly if any collection of pairwise in-tersecting balls has non-empty (coarse) intersection. (
Coarse ) Helly groups are groups actinggeometrically on (coarse) Helly graphs. Our main result is that finitely generated groups thatare hyperbolic relative to (coarse) Helly subgroups are themselves (coarse) Helly. One impor-tant consequence is that various classical groups, including toral relatively hyperbolic groups,are equipped with a CAT(0)-like structure – they act geometrically on spaces with convexgeodesic bicombing. As a means of proving the main theorems we establish a result of inde-pendent interest concerning relatively hyperbolic groups: a ‘relatively hyperbolic’ descriptionof geodesics in a graph on which a relatively hyperbolic group acts geometrically. In the otherdirection, we show that for relatively hyperbolic (coarse) Helly groups their parabolic subgroupsare (coarse) Helly as well. More generally, we show that ‘quasiconvex’ subgroups of (coarse)Helly groups are themselves (coarse) Helly.
Contents
1. Introduction 12. Preliminaries 33. Mapping geodesics to quasi-geodesics 94. Hellyness 165. Quasiconvex subgroups of Helly groups 26References 291.
Introduction
We consider any (simplicial) graph as its vertex set equipped with the combinatorial metric.We say a graph Γ is
Helly (respectively, coarse Helly ) if given any collection of balls { B ρ i ( x i ) | i ∈ I} in Γ such that B ρ i ( x i ) ∩ B ρ j ( x j ) = ∅ for any i, j ∈ I , we have T i ∈I B ρ i ( x i ) = ∅ (respectively, T i ∈I B ρ i + ξ ( x i ) = ∅ for some universal constant ξ ≥ Helly (respectively, coarse Helly ) if it acts geometrically on a Helly (respectively, coarse Helly) graph.See Definition 4.1 for more details.Intuitively, a group G is said to be hyperbolic relative to subgroups H , . . . , H m ≤ G if thesubgroups H j are some (possibly non-hyperbolic) subgroups “arranged in G in a hyperbolicway”. The reader may refer to Section 2.2 for precise definitions, notation and terminology onrelatively hyperbolic groups. Our main results are as follows. Theorem 1.1 (Hellyness of Relatively Hyperbolic Groups) . Let G be a finitely generated groupthat is hyperbolic relative to a collection of Helly subgroups. Then G is Helly. Based on a result of Lang [Lan13] it was shown in [CCG +
20] that all (Gromov) hyperbolicgroups are Helly. As a step towards proving Theorem 1.1 we prove the following ‘coarse’ version.
Theorem 1.2 (Coarse Hellyness of Relatively Hyperbolic Groups) . Let G be a finitely generatedgroup that is hyperbolic relative to a collection of coarse Helly subgroups. Then G is coarse Helly. Mathematics Subject Classification.
Key words and phrases.
Helly groups, coarse Helly groups, relatively hyperbolic groups.The first author was partially supported by (Polish) Narodowe Centrum Nauki, UMO-2017/25/B/ST1/01335.
Helly graphs are classical objects that have been studied intensively within the metric andalgorithmic graph theory for decades. They are also known as absolute retracts (in the categoryof simplicial graphs with simplicial morphisms), and are universal, in the sense that everygraph embeds isometrically into a Helly graph – cf. e.g. the survey [BC08]. The study of groupsacting on such graphs was initiated recently in [CCG + + +
20] is that they act geometrically on injective metric spaces – injective hulls of cor-responding Helly graphs – and hence on spaces with convex geodesic bicombing . The latter aregeodesic spaces in which one has a chosen geodesic between any two points, and the family ofsuch geodesics satisfies some strong convexity properties reminiscent of CAT(0) geodesics. Thisallows one to obtain many CAT(0)-like results for groups acting geometrically on such spaces –see e.g. [DL16]. Theorem 1.1 equips finitely generated groups hyperbolic relative to Helly groupswith such a fine CAT(0)-like structure. As an example, one refines this way the geometry oftoral relatively hyperbolic groups – a classical and widely studied class of groups hyperbolic rel-ative to abelian subgroups (cf. e.g. [DG08]). It is so because all finitely generated abelian groupsand, more generally, all CAT(0) cubical groups are Helly [CCG + +
20, HO19].The notion of coarse Helly graphs has been introduced in [CCG + +
20] – groups acting geometrically on coarse Helly graphs satisfyingan additional condition (of having stable intervals) are Helly – see Theorem 4.2. This way thenotion of coarse Hellyness may be seen as a means for proving Hellyness, and this is exactlywhat is happening in this article – see Section 4. On the other hand, the class of coarse Hellygroups seem to be of interest on its own. Recently it has been shown in [HHP20] that mappingclass groups of surfaces and, more generally, hierarchically hyperbolic groups act geometricallyon coarsely Helly spaces.In the direction converse to Theorems 1.1 & 1.2, we prove the following.
Theorem 1.3 (Parabolic Subgroups of (Coarse) Helly Groups) . Let G be a group hyperbolicrelative to { H , . . . , H m } . If G is Helly then H , . . . , H m are Helly. If G is coarse Helly then H , . . . , H m are coarse Helly. Since (parabolic) subgroups H , . . . , H m as above are (semi-)strongly quasiconvex in the senseof Definition 5.1 (see [DS05]), the result above is an immediate consequence of the followingtheorem concerning arbitrary (coarse) Helly groups. We believe the result is of its own interest,providing further examples of such groups. Theorem 1.4 (Quasiconvex Subgroups of (Coarse) Helly Groups) . Let Γ be a locally finitegraph, let G be a group acting on Γ geometrically, and let H ≤ G be a subgroup. Then thefollowing hold. (i) If Γ is Helly and H is strongly quasiconvex with respect to Γ , then H is Helly. (ii) If Γ is coarse Helly and H is semi-strongly quasiconvex with respect to Γ , then H iscoarse Helly. Our proof of Theorems 1.1 & 1.2 relies on the following result on relatively hyperbolic groups.We believe it is of independent interest from the point of view of the general theory of relativehyperbolicity. In what follows, we let G be a finitely generated group hyperbolic relative tosubgroups H , . . . , H m and let X be a finite generating set for G . For j ∈ { , . . . , m } andan integer N ≥
1, given a proper graph Γ j together with a geometric action H j y Γ j , weconstruct a (Vietoris–Rips) graph Γ j,N by adding edges to Γ j so that v ∼ w in Γ j,N if and onlyif d Γ j ( v, w ) ≤ N . We then construct a graph Γ( N ) by, roughly speaking, taking a disjoint unionof the barycentric subdivision of the Cayley graph Cay( G, X ) and a copy of Γ j,N for each right
ELLY GROUPS, COARSE HELLY GROUPS, AND RELATIVE HYPERBOLICITY 3 coset of H j in G , and adding extra edges to make the graph connected. The graph Γ( N ) thusobtained comes equipped with a geometric action of G . The reader may refer to Section 2.3 forprecise definitions, and to Section 2.4 for a construction of a ‘derived path’ b P ⊆ Cay(
G, X ∪ H )given any path P ⊆ Γ( N ). Theorem 1.5 (Geodesics to Quasi-geodesics) . There exist constants N ≥ , λ ≥ , c ≥ anda finite collection Φ of non-geodesic paths in Γ( N ) with the following property. Let P be a pathin Γ( N ) having no parabolic shortenings and not containing any G -translate of a path in Φ asa subpath. Then the derived path b P in Cay(
G, X ∪ H ) is a -local geodesic ( λ, c ) -quasi-geodesicthat does not backtrack. We prove Theorem 1.5 in Section 3. Theorem 1.2 follows immediately from Proposition 4.5.The latter, together with Theorem 1.1 are proved in Section 4. Theorem 1.4 (implying Theo-rem 1.3) is proved in Section 5.
Acknowledgements.
We thank Oleg Bogopolski for directing us towards Proposition 2.8.2.
Preliminaries
Graphs and hyperbolicity.
In our setting, a graph
Γ is a set V (Γ) of vertices togetherwith a multiset E (Γ) of edges { v, w } for v, w ∈ V (Γ); in particular, we allow loops and multipleedges in a graph. By a path P in a graph Γ we mean a combinatorial path, i.e. a sequence ofvertices v , . . . , v n ∈ V (Γ) and edges { v i , v i +1 } ∈ E (Γ). In this case, | P | := n is said to be the length of P , and we write P − := v and P + := v n for the starting and ending vertices of a path P , respectively. A path of length 0 is said to be trivial . We also write P for the path from P + to P − following the same edges as P just in opposite order. A subpath of P is a path consistingof consecutive edges in P . Furthermore, given paths P , . . . , P k in Γ such that ( P i − ) + = ( P i ) − for 1 ≤ i ≤ k , we write P P · · · P k for the path obtained by concatenating the paths P i .We require all our graphs Γ to be connected, i.e. we assume that for any v, w ∈ V (Γ) thereexists a path P in Γ such that P − = v and P + = w . In this case, we view a graph Γ as a metricspace with underlying set V (Γ) and a metric d Γ ( v, w ) = min {| P | | P is a path in Γ , P − = v, P + = w } . Given some constants λ ≥ c ≥
0, a path P = ( v , . . . , v n ) in a graph Γ is said tobe a ( λ, c ) -quasi-geodesic if | i − j | ≤ λd Γ ( v i , v j ) + c whenever 0 ≤ i, j ≤ n (equivalently, if | Q | ≤ λd Γ ( Q − , Q + ) + c for every subpath Q ⊆ P ). A (1 , geodesic . Given some k ≥
2, we also say a path P is a k -local geodesic (respectively, a k -local ( λ, c ) -quasi-geodesic ) if every subpath Q ⊆ P with | Q | ≤ k is a geodesic (respectively, a( λ, c )-quasi-geodesic).The following definition of hyperbolicity, which we state for graphs, is usually stated forgeodesic metric spaces. However, it is easy to see that a graph Γ is hyperbolic in the sense ofDefinition 2.1 if and only if it is quasi-isometric to a hyperbolic geodesic metric space in theusual sense. Moreover, even though Lemma 2.2 and Theorem 2.3 below are usually stated forgeodesic metric spaces, they can be easily seen to apply to graphs (under our terminology) aswell. Definition 2.1.
Let λ ≥ δ, c ≥
0, and let Γ be a graph. ∆ =
P QR is said to be a geodesic (respectively, ( λ, c ) -quasi-geodesic ) triangle in Γ if P, Q, R ⊆ Γ are geodesic (respectively, ( λ, c )-quasi-geodesic) paths with R + = P − , P + = Q − and Q + = R − . A geodesic triangle ∆ = P QR inΓ is said to be δ -thin if given any two vertices u ∈ R and v ∈ P (respectively, u ∈ P and v ∈ Q , u ∈ Q and v ∈ R ) such that d Γ ( P − , u ) = d Γ ( P − , v ) ≤ | R | + | P |−| Q | (respectively, d Γ ( Q − , u ) = d Γ ( Q − , v ) ≤ | P | + | Q |−| R | , d Γ ( R − , u ) = d Γ ( R − , v ) ≤ | Q | + | R |−| P | ), we have d Γ ( u, v ) ≤ δ . The graphΓ is said to be δ -hyperbolic if all its geodesic triangles are δ -thin; we say that Γ is hyperbolic ifit is δ -hyperbolic for some constant δ ≥ δ -slim . DAMIAN OSAJDA AND MOTIEJUS VALIUNAS
Lemma 2.2 (see [GH90, Proposition 2.21]) . Let Γ be a graph and let δ ≥ . Suppose that forany geodesic triangle ∆ = P QR in Γ and for any vertex u ∈ R , there exists a vertex v ∈ P Q such that d Γ ( u, v ) ≤ δ . Then Γ is hyperbolic. We also use the following well-known result on hyperbolic metric spaces.
Theorem 2.3 (see [GH90, Th´eor`eme 5.11]) . For any δ, c ≥ and λ ≥ , there exists a constant ζ = ζ ( δ, λ, c ) ≥ with the following property. Let Γ be a δ -hyperbolic graph, let P ⊆ Γ be a ( λ, c ) -quasi-geodesic, and let Q ⊆ Γ be a geodesic with Q − = P − and Q + = P + . Then theHausdorff distance between P and Q is at most ζ . A particular case of a graph is the Cayley graph Cay(
G, Y ) of a group G with respect to a(not necessarily finite) generating set Y . Formally, we view Y as an abstract set together witha map ǫ : Y → G such that the image of ǫ generates G : in particular, this allows consideringCayley graphs with multiple edges. We also assume that Y is equipped with an involution ι : Y → Y such that ǫ ( ι ( y )) = ǫ ( y ) − for all y ∈ Y . The Cayley graph Cay( G, Y ) then has G asits vertex set and edge e from g to ǫ ( y ) g for any g ∈ G and y ∈ Y . We label such a directed edge e by the letter y , and identify its inverse e with the edge from ǫ ( y ) g to g labelled by ι ( y ). Forsimplicity of notation, we write d Y ( g, h ) for d Cay(
G,Y ) ( g, h ) whenever g, h ∈ G . Moreover, fora path P ⊆ Cay(
G, Y ) and a symmetric generating set Ω of G , we write | P | Ω for d Ω ( P − , P + );we furthermore allow for the possibility that Ω ⊆ G is an arbitrary symmetric subset of G , inwhich case we set | P | Ω = ∞ whenever ( P + )( P − ) − is not in the subgroup generated by Ω. Remark . All of this formalism regarding the generating sets might seem unnecessary. How-ever, it allows us to use different ‘generators’ to represent the same element of the group G .This allows simplifications in our arguments when G is hyperbolic relative to a collection ofsubgroups { H , . . . , H m } in the case when the subgroups H j intersect non-trivially.We also label any path P in Cay( G, Y ) by a word y · · · y n over Y if P is the path from g to ǫ ( y n ) · · · ǫ ( y ) g (for some g ∈ G ) following the edges labelled by y , . . . , y n , in this order. We say aword over Y is geodesic (respectively, ( λ, c ) -quasi-geodesic ) if it labels a geodesic (respectively,( λ, c )-quasi-geodesic) path P ⊆ Cay(
G, Y ); note that this property does not depend on thechoice of P . We similarly define k -local geodesic and k -local ( λ, c )-quasi-geodesic words over Y .2.2. Relatively hyperbolic groups.
Our approach to relatively hyperbolic groups follows theapproach of D. V. Osin [Osi06].Let G be a group and let H , . . . , H m be a finite collection of distinct subgroups of G . Suppose G is finitely generated, i.e. there exists a surjective group homomorphism ˆ ǫ : F ( X ) → G for a finite set X ; without loss of generality, suppose that ˆ ǫ | X is injective, and that ˆ ǫ ( X )is symmetric and does not contain 1 ∈ G . Thus, ˆ ǫ extends to a surjective homomorphism ǫ : F = F ( X ) ∗ ( ∗ mi =1 e H i ) → G , such that ǫ maps each group e H i isomorphically onto H i . We saythat G is finitely presented with respect to { H , . . . , H m } if ker ǫ is the normal closure (in F ) ofa finite subset R ⊆ F . We also write(1) G = h X, { H , . . . , H m } | Ri for a relative presentation of G , which is said to be finite if X and R are finite.Now suppose G is finitely presented with respect to { H , . . . , H m } with a finite relativepresentation (1). Let H = F mj =1 ( e H j \ { } ). We say f : N → N is a relative isoperimetricfunction of the presentation (1) if for all n ≥ W over X ∪ H of length n with ǫ ( W ) = G W = F k Y i =1 g − i R ± i g i for some k ≤ f ( n ) and some elements g i ∈ F , R i ∈ R . A minimal relative isoperimetric function(if it exists) is called a relative Dehn function , and a function f : N → N is said to be linear ifthere exist a, b ∈ N such that f ( n ) ≤ an + b for all n . ELLY GROUPS, COARSE HELLY GROUPS, AND RELATIVE HYPERBOLICITY 5
Definition 2.5 (Osin [Osi06, Definition 2.35]) . We say that the group G is hyperbolic relativeto { H , . . . , H m } if G is finitely presented with respect to { H , . . . , H m } , and the relative Dehnfunction of this presentation exists and is linear.Now consider the Cayley graph Cay( G, X ∪ H ), defined in Section 2.1 for Y = X ∪ H . In thiscase, we take the involution ι : X ∪ H → X ∪ H such that ι ( X ) = X and ι ( e H j \ { } ) = e H j \ { } for each j : together with the condition that ǫ ( ι ( y )) = ǫ ( y ) − for each y ∈ X ∪ H , this defines ι uniquely.We say a path P in Cay( G, X ∪ H ) is an H j -path if all edges of P are labelled by elementsof e H j ; an H j -path of length 1 is called an H j -edge or an H -edge . A maximal H j -subpath of apath P is said to be an H j -component (or simply a component ). Given two paths P and Q inCay( G, X ∪ H ), an H i -component P ′ of P is said to be connected to an H j -component Q ′ of Q if i = j and P ′− ( Q ′− ) − ∈ H j (note that this defines an equivalence relation). A component P ′ of a path P is said to be isolated if it is not connected to any other component of P , and wesay a path P in Cay( G, X ∪ H ) does not backtrack if all its components are isolated. We alsosay that a subword of a word P over X ∪ H is a component of P if the corresponding subpathof a path P labelled by P is a component of P , and that P does not backtrack if P does notbacktrack.A vertex v of a path P is said to be non-phase if it belongs to the interior of some componentof P , and phase otherwise. Note that a geodesic path does not backtrack and all its verticesare phase. Given k ∈ N , two paths P , Q are said to be k -similar if d X ( P − , Q − ) ≤ k and d X ( P + , Q + ) ≤ k . For future reference, given a path P = P UP in Cay( G, X ∪ H ) we define V to be a subpath of P preceding U if V is a subpath of P .In our proofs of Theorem 1.5 and Proposition 2.10 below, we use the following results dueto D. V. Osin. In all of these results, G is a (fixed) finitely generated group that is hyperbolicrelative to subgroups { H , . . . , H m } , X is a finite generating set for G , and H is as above. Theorem 2.6 (Bounded Coset Penetration; Osin [Osi06, Theorem 3.23]) . For any λ ≥ , c ≥ and k ∈ N , there exists a constant ε = ε ( λ, c, k ) ∈ N with the following property. Let P and Q be two k -similar ( λ, c ) -quasi-geodesics in Cay(
G, X ∪ H ) that do not backtrack. Then (i) for any phase vertex u of Q , there exists a phase vertex v of P such that d X ( u, v ) ≤ ε ; (ii) for any component Q ′ of Q with d X ( Q ′− , Q ′ + ) > ε , there exists a component of P con-nected to Q ′ ; and (iii) for any two connected components P ′ , Q ′ of P , Q (respectively), we have d X ( P ′− , Q ′− ) ≤ ε and d X ( P ′ + , Q ′ + ) ≤ ε . Theorem 2.7 (Osin [Osi06, Theorem 3.26]) . There exists a constant ν ∈ N such that thefollowing holds. Let ∆ = PQR be a geodesic triangle in
Cay(
G, X ∪ H ) . Then for any vertex u of P , there exists a vertex v of Q ∪ R such that d X ( u, v ) ≤ ν . Proposition 2.8 (Osin [Osi07, Proposition 3.2]) . For any λ ≥ and c ≥ , there exists afinite subset Ω ⊆ G and a constant L = L ( λ, c ) ∈ N such that the following holds. Let n ≥ ,let Q = P · · · P n be a closed path in Cay(
G, X ∪ H ) , and suppose that there exists a subset I ⊆ { , . . . , n } such that P i is an isolated component of Q if i ∈ I and a ( λ, c ) -quasi-geodesicotherwise. Then the Ω -lengths of the P i for i ∈ I satisfy X i ∈ I | P i | Ω ≤ Ln.
Remark . Theorems 2.6 and 2.7 are stated in [Osi06] only for generating sets X ⊆ G thatsatisfy a certain technical condition stated in the beginning of [Osi06, § X (up to possibly increasing the constant ε ). Using Theorem 2.6(i), we can alsoshow that Theorem 2.7 holds independently of the choice of a finite symmetric generating set X . DAMIAN OSAJDA AND MOTIEJUS VALIUNAS
We end our introduction to relatively hyperbolic groups by proving the following result, whichwe use in our proof of Theorem 1.2. This can be viewed as a version of Theorem 2.6 stated fortriangles instead of ‘bigons’.
Proposition 2.10.
For any λ ≥ and c ≥ , there exists a constant µ = µ ( λ, c ) with thefollowing property. Let ∆ = PQR be a non-backtracking ( λ, c ) -quasi-geodesic triangle in theCayley graph Cay(
G, X ∪ H ) (i.e. P , Q , R are ( λ, c ) -quasi-geodesics in Cay(
G, X ∪ H ) that donot backtrack such that P + = Q − , Q + = R − and R + = P − ). Then (i) for any phase vertex u of R , there exists a phase vertex v of P or of Q such that d X ( u, v ) ≤ µ ; (ii) for any component R ′ of R with d X ( R ′− , R ′ + ) > µ , there exists a component of P or of Q connected to R ′ ; (iii) if a component R ′ of R is connected to a component P ′ of P but is not connected to anycomponent of Q , we have d X ( R ′ + , P ′− ) ≤ µ and d X ( P ′ + , R ′− ) ≤ µ ; and (iv) if a component R ′ of R is connected to a component P ′ of P and a component Q ′ of Q ,then d X ( R ′ + , P ′− ) ≤ µ , d X ( P ′ + , Q ′− ) ≤ µ and d X ( Q ′ + , R ′− ) ≤ µ . R Q P R QP uu v v (a) Part (i). R R ′ R QP (b) Part (ii). R R ′ R P P ′ P Qe e (c) Part (iii). R R ′ R P P ′ P Q Q ′ Q e e e (d) Part (iv). Figure 1.
The proof of Proposition 2.10.
Proof.
Let ε = ε ( λ, c, ∈ N be as in Theorem 2.6, let ν ∈ N be as in Theorem 2.7, and let Ω ⊆ G and L = L ( λ, c ) ∈ N be as in Proposition 2.8. Since | Ω | < ∞ , we have M := sup ω ∈ Ω | ω | X < ∞ ;it follows that if d Ω ( g, h ) ≤ D for some g, h ∈ G and D ∈ N then d X ( g, h ) ≤ DM . We set µ := max { ε + ν, M L } .We now prove parts (i)–(iv) in the statement of the theorem.(i) Let P , Q , R ⊆ Cay(
G, X ∪ H ) be geodesics such that ( P ) + = P + = Q − = ( Q ) − ,( Q ) + = Q + = R − = ( R ) − and ( R ) + = R + = P − = ( P ) − , so that ∆ ′ = P Q R is ageodesic triangle in Cay( G, X ∪ H ); see Figure 1(a). Thus the paths P , Q , R , P , Q , R are all ( λ, c )-quasi-geodesics that do not backtrack.Let u ∈ R be a phase vertex. By Theorem 2.6(i), there is a phase vertex u ∈ R such that d X ( u, u ) ≤ ε . By Theorem 2.7, there is a vertex v or either P or Q suchthat d X ( u , v ) ≤ ν ; note that since P and Q are geodesics, v is necessarily a phasevertex. Finally, by Theorem 2.6(i) again, there exists a phase vertex v of P (if v ∈ P )or of Q (if v ∈ Q ) such that d X ( v , v ) ≤ ε . We thus have d X ( u, v ) ≤ d X ( u, u ) + d X ( u , v ) + d X ( v , v ) ≤ ε + ν ≤ µ, as required.(ii) Suppose that R ′ is an isolated component of PQR . We can then write R = R R ′ R ,so that P , Q , R , R are all ( λ, c )-quasi-geodesics, and R ′ is an isolated component of PQR R ′ R : see Figure 1(b). It follows from Proposition 2.8 that | R ′ | Ω ≤ L , and so | R ′ | X ≤ M L . This contradicts the fact that d X ( R ′− , R ′ + ) > µ .Thus R ′ must be connected to some other component of PQR . As R does not back-track, it follows that R ′ is connected to a component of either P or Q , as required. ELLY GROUPS, COARSE HELLY GROUPS, AND RELATIVE HYPERBOLICITY 7 (iii) Let j ∈ { , . . . , m } be such that P ′ and R ′ are H j -components. Thus there exist H j -edges e and e such that ( e ) − = P ′− , ( e ) + = R ′ + , ( e ) − = R ′− and ( e ) + = P ′ + ; moreover,we may write P = P P ′ P and R = R R ′ R : see Figure 1(c). Since P and R do notbacktrack, and since R ′ (and therefore e ) is not connected to any component of Q ,it follows that e and e are isolated components of R P e and P QR e , respectively.Moreover, R , P , P , Q and R are all ( λ, c )-quasi-geodesics.It follows from Proposition 2.8 that | e | Ω ≤ L and | e | Ω ≤ L ; therefore, we have | e | X ≤ M L ≤ µ and | e | X ≤ M L ≤ µ . Hence d X ( P ′− , R ′ + ) , d X ( R ′− , P ′ + ) ≤ µ , asrequired.(iv) Let j ∈ { , . . . , m } be such that P ′ , Q ′ and R ′ are H j -components. Thus there exist H j -edges e , e and e such that ( e ) − = P ′− , ( e ) + = R ′ + , ( e ) − = Q ′− , ( e ) + = P ′ + ,( e ) − = R ′− , ( e ) + = Q ′ + ; moreover, we may write P = P P ′ P , Q = Q Q ′ Q and R = R R ′ R : see Figure 1(d). Since P , Q and R do not backtrack, it follows that e , e and e are isolated components of R P e , P Q e and Q R e , respectively. Moreover, R , P , P , Q , Q and R are all ( λ, c )-quasi-geodesics.It follows from Proposition 2.8 that | e i | Ω ≤ L , and so | e i | X ≤ M L ≤ µ , for each i ∈ { , , } . Hence d X ( P ′− , R ′ + ), d X ( Q ′− , P ′ + ) and d X ( R ′− , Q ′ + ) are all bounded above by µ , as required. (cid:3) The graph Γ( N ) . We now construct an action of G on a graph Γ( N ) given an action of e H j ≤ G on a graph Γ j for each j ∈ { , . . . , m } , so that if the actions e H j y Γ j are all geometricthen so is G y Γ( N ). Roughly speaking, we take the barycentric subdivision of the Cayleygraph Cay( G, X ) together with a copy of a graph Γ j,N (obtained by adding extra edges to Γ j )for each right coset of H j in G , and glue them together using ‘connecting edges’ in a consistentway.Thus, let G be a group with a finite symmetric generating set X , let H , . . . , H m ≤ G bea collection of subgroups, and, for each j , let e H j be an isomorphic copy of H j acting on asimplicial graph Γ j by isometries, and fix a vertex v j ∈ Γ j . Let F = F ( X ) ∗ ( ∗ mj =1 e H j ), let H = F mj =1 ( e H j \ { } ), let ǫ : F → G be the canonical surjection, and let ι : X → X be aninvolution such that ǫ ( ι ( x )) = ǫ ( x ) − for all x ∈ X . For each j , let π j : F → e H j be thecanonical retraction, defined as the identity map on e H j and as the trivial map on F ( X ) and on e H i for i = j .We now construct a graph e Γ( N ) by taking a copy of the barycentric subdivision of the Cayleygraph Cay( F ( X ) , X ) for each right coset of F ( X ) in F and a copy of Γ j,N for each right cosetof e H j in F , and connecting them using auxiliary edges. Definition 2.11.
Let N ≥
1. We construct a simplicial graph e Γ( N ) as follows.(i) For each j ∈ { , . . . , m } , define a graph Γ j,N with vertices V (Γ j,N ) = V (Γ j ) and edges { v, w } whenever d Γ j ( v, w ) ≤ N .(ii) The vertices of e Γ( N ) are V ( e Γ( N )) = e V free ⊔ e V med ⊔ e V int , where(a) e V free = F , the free vertices ;(b) e V med = ( F × X ) / ∼ , where ( g, x ) ∼ ( h, y ) if and only if ( h, y ) ∈ { ( g, x ) , ( xg, ι ( x )) } ,the medial vertices ; and(c) e V int = F mj =1 ( e H j \ F ) × V (Γ j ), where e H j \ F is the set of right cosets of e H j in F , the internal vertices .(iii) The edges of e Γ( N ) are of three types:(a) the free edges : { g, [( g, x )] } for vertices g ∈ e V free and [( g, x )] ∈ e V med ;(b) the connecting edges : n g, ( e H j g, v j · π j ( g )) o for g ∈ e V free and ( e H j g, v j · π j ( g )) ∈ e V int ;and(c) the internal edges : n ( e H j g, v ) , ( e H j g, w ) o for vertices ( e H j g, v ) , ( e H j g, w ) ∈ e V int suchthat { v, w } ∈ E (Γ j,N ). DAMIAN OSAJDA AND MOTIEJUS VALIUNAS
We define a right action of F on V ( e Γ( N )) as follows: for g ∈ F , h · g = hg for h ∈ V free ;[( h, x )] · g = [( hg, x )] for [( h, x )] ∈ V med ; and( e H j h, v ) · g = ( e H j hg, v · π j ( g )) for ( e H j h, v ) ∈ V int . It is easy to see that this is indeed a well-defined action, and that it sends edges of e Γ( N ) toedges, thus inducing an action of F on e Γ( N ). It is also clear that this action preserves the typesof vertices (free, medial or internal) and edges (free, connecting or internal): see Figure 2.Γ ,N / e H Γ m,N / e H m Figure 2.
The quotient graph e Γ( N ) /F . Orbits of medial vertices are shownin green, connecting edges in blue, free vertices and edges in gray. Orbits ofinternal vertices and edges are represented by the red regions.Finally, we define the graph Γ( N ) as the quotient Γ( N ) = e Γ( N ) / ker( ǫ ), so that we have anaction G y Γ( N ) induced by F y e Γ( N ). The following result is straightforward. Lemma 2.12 (Description of Γ( N )) . We have V (Γ( N )) = V free ⊔ V med ⊔ V int , where V free = G are the free vertices, V med = ( G × X ) / ∼ are the medial vertices, where ( g, x ) ∼ ( h, y ) if andonly if ( h, y ) ∈ { ( g, x ) , ( ǫ ( x ) g, ι ( x )) } , and V int = F mj =1 ( H j \ G ) × V (Γ j ) are the internal vertices.The edges of Γ( N ) can be partitioned intofree: { g, [( g, x )] } for g ∈ V free and [( g, x )] ∈ V med , connecting: { g, ( H j g, u j,g ) } for g ∈ V free , ( H j g, u j,g ) ∈ V int and some u j,g ∈ V (Γ j ) , internal: { ( H j g, v ) , ( H j g, w ) } for ( H j g, v ) , ( H j g, w ) ∈ V int with { v, w } ∈ E (Γ j,N ) . (cid:3) In particular, it follows that for any j ∈ { , . . . , m } and any H j g ∈ H j \ G , the subset { ( H j g, v ) | v ∈ V (Γ j ) } ⊂ V (Γ( N )) spans a subgraph isomorphic to Γ j,N . We will refer tothis subgraph as the g -copy (or just a copy ) of Γ j,N , and we say that a path P penetrates acopy Γ of Γ j,N if P ∩ Γ = ∅ .Given a path P ⊆ Γ( N ), we also say that P has no parabolic shortenings if every subpathof P all of whose vertices are internal – that is, a subpath contained in some copy Γ ⊆ Γ( N )of Γ j,N – is a geodesic when viewed as a path in Γ . (This terminology is taken from [AC16],which in turn arises from the notion of parabolic subgroups: when G is hyperbolic relative to { H , . . . , H m } , a subgroup of G is said to be parabolic if it is conjugate to some H j .) Lemma 2.13 (Properties of G y Γ( N )) . Suppose that, for each j ∈ { , . . . , m } , the graph Γ j is proper (as a metric space). Then the graphs e Γ( N ) and Γ( N ) are proper, and the followinghold. (i) If each action e H j y Γ j is properly discontinuous, then so are the actions F y e Γ( N ) and G y Γ( N ) . (ii) If each action e H j y Γ j is cocompact, then so are the actions F y e Γ( N ) and G y Γ( N ) .Proof. To show that e Γ( N ) is proper, it is enough to show that each vertex of e Γ( N ) is incident tofinitely many edges. But a free vertex of e Γ( N ) is incident to | X | + m < ∞ edges, a medial vertex ELLY GROUPS, COARSE HELLY GROUPS, AND RELATIVE HYPERBOLICITY 9 is incident to two edges, and an e H j -internal vertex ( e H j g, u ) is incident to at most d j,N ( u ) + | Stab e H j ( u ) | < ∞ edges, where d j,N ( u ) is the number of vertices v ∈ Γ j such that d Γ j ( u, v ) ≤ N .Thus e Γ( N ) is proper as a metric space. As Γ( N ) is a quotient of e Γ( N ), it follows that Γ( N ) isproper as well.We now prove (i) and (ii).(i) Since e H j y Γ j is properly discontinuous, each vertex of Γ j has a finite e H j -stabiliser.Now each free vertex and each medial vertex of e Γ( N ) has a trivial stabiliser. Moreover, itis easy to check that Stab F ( e H j g, u ) = g − h Stab e H j (cid:0) u · π j ( g ) − (cid:1)i g for any e H j -internalvertex ( e H j g, u ), and so | Stab F ( e H j g, u ) | < ∞ . Thus vertices of e Γ( N ) have finite F -stabilisers and so the action of F on e Γ( N ) is properly discontinuous. Since G -stabilisersof vertices in Γ( N ) are just images of F -stabilisers of vertices in e Γ( N ) under ǫ , it followsthat they are finite, and so G y Γ( N ) is properly discontinuous as well.(ii) Since Γ j is proper and the action of e H j on Γ j is cocompact (for each j ), the action of e H j on Γ j,N is cocompact as well. It is then easy to check that e Γ( N ) has m < ∞ orbitsof connecting edges, 2 · | X/ ∼| < ∞ orbits of free edges, and P mj =1 (cid:12)(cid:12)(cid:12) E (Γ j,N / e H j ) (cid:12)(cid:12)(cid:12) < ∞ orbits of internal edges (see also Figure 2). Thus the action of F on e Γ( N ) is cocompact,and since e Γ( N ) /F ∼ = Γ( N ) /G so is the action of G on Γ( N ). (cid:3) Derived paths.
Here we define a derived path b P in Cay( G, X ∪ H ) corresponding to agiven path P in Γ( N ). Our definition is an adaptation of a construction of Y. Antol´ın andL. Ciobanu [AC16, Construction 4.1]. Definition 2.14.
Let N ≥
1. For each vertex v ∈ V (Γ( N )), we define a path Z v ⊆ Γ( N ) suchthat ( Z v ) − = v and ( Z v ) + is a free vertex, and such that | Z v | is as small as possible under theseconditions. Given a path P ⊆ Γ( N ), we define the derived path b P ⊆ Cay(
G, X ∪ H ) in thefollowing steps.(i) If P has no free vertices, then set n = 1 and P = Z P − P Z P + , and proceed to step (iii).Otherwise, write P = P ′ P · · · P n − P ′′ in such a way that P ′ + = ( P ) − , ( P ) − = ( P ) + ,. . . , ( P n − ) + = P ′′− are all free, no other vertices of P are free, and | P i | ≥ ≤ i ≤ n − | P ′ | ≤
3, then set P to be the trivial path with ( P ) − = ( P ) + = P ′ + ; otherwise, set P = Z P − P ′ . Similarly, if | P ′′ | ≤
3, then set P n to be the trivial path with ( P n ) − =( P n ) + = P ′′− ; otherwise, set P n = P ′′ Z P + . If | P ′ | ≥ | P ′′ | ≥ P (respectively, P n ) an extended subpath of P .It now follows from the construction that for 1 ≤ i ≤ n , the vertices ( P i ) − and ( P i ) + are the only free vertices of P i .(iii) Let b P = b P b P · · · b P n be the path in Cay( G, X ∪ H ) such that, for 1 ≤ i ≤ n , b P i is a pathof length ≤ b P i ) − = ( P i ) − and ( b P i ) + = ( P i ) + , as follows.(a) If ( P i ) − = ( P i ) + , then we set b P i to be an empty path.(b) If ( P i ) − = ( P i ) + and all non-endpoint vertices of P i are medial, then | P i | = 2 and( P i ) + = ǫ ( x )( P i ) − for some x ∈ X ; we then set b P i to be an edge labelled by x .(c) Otherwise, all non-endpoint vertices of P i are H j -internal for some j , and ( P i ) + = ǫ ( h )( P i ) − for some h ∈ e H j ; we then set b P i to be an e H j -edge labelled by h .An example construction of b P is shown in Figure 3.3. Mapping geodesics to quasi-geodesics
In this section we prove Theorem 1.5. In order to do this, we proceed in two steps. We firstshow that the word labelling a 2-local geodesic path in Cay(
G, X ∪H ), that does not contain anyof the finitely many ‘strongly non-geodesic’ prohibited subwords, labels a ( λ, c )-quasi-geodesicthat does not backtrack (for some fixed λ ≥ c ≥ PZ P − P P P n − P ′ P ′′ b P b P b P n − Figure 3.
An example construction of a derived path b P ⊆ Cay(
G, X ∪ H ) givena path P ⊆ Γ( N ). In this case, | P ′ | ≥ | P ′′ | ≤
3, and the paths P n and b P n aretrivial.that for N big enough, if P is a 5-local geodesic in Γ( N ) with no parabolic shortenings and withno subpaths Q such that b Q is labelled by one of the aforementioned prohibited words, then b P satisfies the premise of Proposition 3.1.3.1. Local geodesics are quasi-geodesics.
We start by analysing 2-local geodesics in thegraph Cay(
G, X ∪ H ). The following result is a strengthening of a result of Y. Antol´ın andL. Ciobanu [AC16, Theorem 5.2], and the proof given in [AC16] carries through to prove thefollowing Proposition after several minor modifications. Here, we say a word P over X ∪ H vertex backtracks if P contains a subword Q representing an element of H j (for some j ) with |Q| ≥
2. Clearly, a word that does not vertex backtrack does not backtrack either.
Proposition 3.1.
There exist constants λ ≥ , c ≥ , and a finite collection b Φ of words over X ∪ H labelling paths Q ⊆ Cay(
G, X ∪ H ) with | Q | > d X ∪H ( Q − , Q + ) such that the followingholds. Let P be a -local geodesic word in Cay(
G, X ∪ H ) that does not contain any element of b Φ as a subword. Then P is a ( λ, c ) -quasi-geodesic word that does not vertex backtrack. R ′ Q ′ P ′ Q ′′ (a) A path P that vertex backtracks. Q Q Q n P P P n U T U T U n − T n S S S n − S n R R R n − R n (b) A non-(4 , P and a geodesic Q . Figure 4.
The proof of Proposition 3.1.
Proof.
We use the following ‘local to global’ property of quasi-geodesics for a hyperbolic spaces[CDP90, Chapter 3, Theorem 1.4]: given a hyperbolic metric space Y and constants λ ′ ≥ c ′ ≥
0, there exist constants λ ≥ c, k ≥ k -local ( λ ′ , c ′ )-quasi-geodesic in Y is a ( λ, c )-quasi-geodesic. In particular, since Cay( G, X ∪ H ) is hyperbolic (seeProposition 2.10(i)), we may use this property for λ ′ = 4 and c ′ = 3: there exist constants λ ≥ c, k ≥ k -local (4 , G, X ∪ H ) is a ( λ, c )-quasi-geodesic. We may further increase k if necessary to assume that k ≥ λ + c .Now let b Ψ be the set of cyclic paths Q in Cay( G, X ∪ H ) that do not backtrack with | Q | < k .By Proposition 2.8, there exists a finite subset Ω ⊆ G and a constant L ≥ Q ∈ b Ψ we have n ( Q ) X i =1 | P Q ,i | Ω < kL, where P Q , , . . . , P Q ,n ( Q ) is the set of words labelling the components of Q . In particular, as X and Ω are finite, it follows that the set of cyclic words labelling elements of b Ψ is finite.
ELLY GROUPS, COARSE HELLY GROUPS, AND RELATIVE HYPERBOLICITY 11
We then define b Φ to be the set of words labelling a path Q in Cay( G, X ∪H ) such that QP ∈ b Ψ(in particular, Q − = P + and Q + = P − ) and | Q | > | P | for some path P in Cay( G, X ∪ H ). Itis clear that b Φ is finite, and that every word in b Φ labels a path Q with | Q | > d X ∪H ( Q − , Q + ).Now let P be a 2-local geodesic word over X ∪ H that does not contain any element of b Φ asa subword. We claim that P is a k -local (4 , P is a k -local (4 , |P| ≤ k . Let P ⊆ Cay(
G, X ∪ H ) be a path labelled by P .We first claim that P does not vertex backtrack. Indeed, if it did, then we would have P = Q ′ P ′ Q ′′ where P ′ labels an element of H j (for some j ) and | P ′ | ≥
2. Without loss ofgenerality, suppose that P ′ is a minimal subpath with this property, and let R ′ be a geodesicpath with R ′± = P ′± (so that either | R ′ | = 1 and R ′ is an e H j -edge, or | R ′ | = 0): see Figure 4(a).As P is a 2-local geodesic, we have | P ′ | ≥
3; by minimality of P ′ , no two components of P ′ are connected, and no component of P ′ is connected to the H j -component R ′ (if | R ′ | = 1). As3 ≤ | P ′ | ≤ | P | ≤ k and so | R ′ | ≤ < ≤ k , it follows that | P ′ R ′ | < k and so P ′ R ′ ∈ b Ψ. Butas | R ′ | ≤
1, we have | P ′ | ≥ > ≥ | R ′ | and so P ′ is labelled by a word in b Φ, contradictingthe choice of P .Thus, P is a 2-local geodesic path of length | P | ≤ k that does not vertex backtrack. Supposefor contradiction that P is not a (4 , P if necessary,we may assume that there exists a geodesic path Q in Cay( G, X ∪ H ) such that Q − = P − , Q + = P + , and such that | P | > | Q | + 3. As P does not vertex backtrack and as Q is a geodesic,both P and Q do not backtrack. Also, as P and Q are 2-local geodesics, all components of P and of Q are edges.Now for some n ≥
0, we may write P = R P R · · · P n R n and Q = S Q S · · · Q n S n in sucha way that, for each i , P i is a component of P connected to a component Q i of Q , and nocomponent of R i is connected to a component of S i . Let T i (respectively, U i ) be the geodesicin Cay( G, X ∪ H ) from ( R i ) − to ( S i ) − (respectively, from ( R i ) + to ( S i ) + ), so that either T i (respectively, U i ) has length 0, or it is an H -edge that is connected, as a component, to P i and Q i (respectively, P i +1 and Q i +1 ); see Figure 4(b).If we had | R i | ≤ ( | S i | + 2) for each i , then we would have | P | = n + n X i =0 | R i | ≤ n + 3 + 32 n X i =0 | S i | ≤ n + n X i =0 | S i | ! + 3 = 4 | Q | + 3 , contradicting the assumption that | P | > | Q | + 3. Thus, there exists some i ∈ { , . . . , n } , whichwe fix from now on, such that | R i | > ( | S i | + 2).Now if | T i | = 1 (as opposed to | T i | = 0), since T i is connected to a component P i of P and since P does not backtrack, it follows that T i is not connected to any component of R i .Similarly, if | T i | = 1 then T i is not connected to any component of S i , and if | U i | = 1 then U i is not connected to any component of R i or of S i . Since P and Q do not backtrack, no twocomponents of R i and no two components of S i are connected. By construction, no componentof R i is connected to a component of S i , and if T i was a component connected to a component U i then we would have | R i | ≤ P does not vertex backtrack), contradicting the fact that | R i | > ( | S i | + 2) ≥ R i U i S i T i are isolated. Moreover, since k ≥ | P | ≥ | R i | > ( | S i | + 2) we have | R i U i S i T i | = | R i | + ( | S i | + 2) < k . Thus R i U i S i T i ∈ b Ψ,and as | R i | > ( | S i | + 2) ≥ | U i S i T i | , the path R i is labelled by a word in b Φ, contradicting thechoice of P .Thus, if P is a 2-local geodesic word in Cay( G, X ∪ H ) that does not contain any element of b Φ as a subword, then P is a k -local (4 , P of length ≤ k does not vertex backtrack. In particular, P is a ( λ, c )-quasi-geodesic. If P did vertexbacktrack then we would have P = Q ′ P ′ Q ′′ where P ′ represents an element of some H j and |P ′ | > k (cf Figure 4(a)); but k was chosen so that k ≥ λ + c , so this is impossible since P is a ( λ, c )-quasi-geodesic. Hence P is a ( λ, c )-quasi-geodesic word that does not vertex backtrack,as required. (cid:3) Local geodesics map to local geodesics.
Throughout the remainder of this section,we adopt the following assumption.
Assumption 3.2.
We assume that each Γ j is proper (as a metric space), and e H j acts on Γ j geometrically. We furthermore assume that N ≥ j / e H j between v j · e H j and any other vertex is at most N .(ii) For each word P ∈ b Φ, where b Φ is the finite collection of words in Proposition 3.1, fixa geodesic word U P over X ∪ H such that U P and P represent the same element of G .Then for any j and any e H j -letter h of U P , we have d Γ j ( v j , v j · h ) ≤ N .(iii) If P is a cyclic word over X ∪ H of length |P| ≤ h is an e H j -letter of P , then d Γ j ( v j , v j · h ) ≤ N .Note that, under the assumption that the actions e H j y Γ j are all geometric, the points(i)–(iii) in Assumption 3.2 will all be satisfied whenever N ≥ e H j acts on Γ j geometrically (and so cocompactly), the graph Γ j / e H j is finite, and so (i) issatisfied for N large enough. Furthermore, as the collection b Φ in Proposition 3.1 is finite, (ii)is true when N is large enough. Finally, it follows from Proposition 2.8 that there are finitelymany cyclic words over X ∪ H of length ≤ N . In fact, one can see from the proof ofProposition 3.1 that (ii) implies (iii) – but both points are stated here for future reference.In the following two lemmas, let P be a path in Γ( N ) that has no parabolic shortenings andat least one free vertex, and let P , . . . , P n ⊆ Γ( N ) and b P = b P · · · b P n ⊆ Cay(
G, X ∪ H ) be asin Definition 2.14. e P ′ e ′ g ( H j g, u ) ǫ ( h ) g ( H j g, w ) (a) Lemma 3.3: the path P . Te T e T P − (b) Lemma 3.4: the paths Z P − (green) and P ′ (blue). Figure 5.
The proofs of Lemmas 3.3 and 3.4. The red shaded area representsa copy of Γ j,N in Γ( N ). Lemma 3.3.
Suppose that P is a closed path, that P − = P + is a free vertex, and that | b P | ≤ .If, for some i , b P i is an isolated component of b P , then | P i | ≤ .Proof. Note that since P − and P + are free vertices, it follows from the construction in Defini-tion 2.14 that P = P · · · P n − , and that P and P n are empty paths. Suppose that b P i is anisolated component of b P : without loss of generality, i = 2. Then the non-endpoint vertices of P are either all H j -internal (for some j ) or all medial. In the latter case, we have | P | = 2and we are done. Thus, without loss of generality, suppose all non-endpoint vertices of P are H j -internal, and so b P is labelled by an element h ∈ e H j .Now it follows from Assumption 3.2(iii) that d Γ j ( v j , v j · h ) ≤ N . By construction, it alsofollows that P = eP ′ e ′ , where e and e ′ are the connecting edges with e − = g , e ′− = ǫ ( h ) g , e + = ( H j , v j ) · g = ( H j g, u ) and e ′ + = ( H j , v j · h ) · g = ( H j g, w ) for some g ∈ G and some u, w ∈ V (Γ j ) with d Γ j ( u, w ) = d Γ j ( v j , v j · h ) ≤ N , and all vertices of P ′ are H j -internal; seeFigure 5(a). Since P has no parabolic shortenings, it follows that P ′ is a geodesic as a path inthe g -copy of Γ j,N , and so | P ′ | = d Γ j,N ( u, w ) = (cid:24) d Γ j ( u, w ) N (cid:25) ≤ . ELLY GROUPS, COARSE HELLY GROUPS, AND RELATIVE HYPERBOLICITY 13
Thus | P | = 2 + | P ′ | ≤
3, as required. (cid:3)
Lemma 3.4.
If, for some i ∈ { , n } , P i is an extended subpath of P , then b P i is an e H j -edge(for some j ) and, for any path Q in Γ( N ) with Q − = ( P i ) − , Q + = ( P i ) + and all other verticesof Q being H j -internal, we have | Q | ≥ .Proof. Suppose that i = 1, and so P is an extended subpath of P : the case i = n is similar.By the construction described in Definition 2.14(ii), we have P = Z P − P ′ , where P ′ is an initialsubpath of P with | P ′ | ≥
4. Here, Z P − is a shortest path in Γ( N ) with ( Z P − ) − = P − and( Z P − ) + a free vertex, and so it follows from Assumption 3.2(i) that | Z P − | ≤ Z P − = T e and P ′ = T e , where e and e are connecting edgesand all edges on the path T T are H j -internal; see Figure 5(b). Since | T | − | T | ≥ − P (and so its subpath T ) has no parabolic shortenings, it follows that any path between( T ) + and ( T ) + consisting only of internal edges has length ≥
2. But a path Q satisfying theconditions stated must be of the form Q = e T e , where T is a path between ( T ) + and ( T ) + consisting only of internal edges, and so we have | Q | = 2 + | T | ≥
4, as required. (cid:3)
We now show that local geodesics in Γ( N ) with no parabolic shortenings are transformed tolocal geodesics in Cay( G, X ∪ H ). Proposition 3.5.
Let P be a -local geodesic in Γ( N ) which has no parabolic shortenings. Thenthe derived path b P is a -local geodesic in Cay(
G, X ∪ H ) . R ′ P ′ i P ′ i +1 P i P i +1 (a) General setup. e T e T e (b) b P ′ i , b P ′ i +1 in the same component. e T P ′ i e T e T (c) b P ′ i +1 , R ′ in the same component. Figure 6.
The proof of Proposition 3.5.
Proof.
The conclusion is trivial if | b P | ≤
1: we will thus assume that | b P | ≥
2, and so P hasat least one free vertex. Let P , . . . , P n ⊆ Γ( N ) and b P = b P · · · b P n ⊆ Cay(
G, X ∪ H ) be as inDefinition 2.14. Note that by the construction we have | b P i | ≤ ≤ i ≤ n .Notice first that for 2 ≤ i ≤ n −
1, we have ( P i ) − = ( P i ) + . Indeed, by construction we have | P i | ≥
1, and ( P i ) − and ( P i ) + are the only free vertices of P i . If we had ( P i ) − = ( P i ) + , then,since P has no parabolic shortenings, it would follow that P i = ee for some connecting edge e ,contradicting the fact that P is 2-local geodesic. Thus we have ( P i ) − = ( P i ) + , and so | b P i | = 1,for 2 ≤ i ≤ n −
1. It follows, in particular, that any subpath of b P of length 2 is of the form b P i b P i +1 for some i ∈ { , . . . , n − } .Now suppose for contradiction that b P is not a 2-local geodesic, and so there exists an i suchthat | b P i | = | b P i +1 | = 1 and d X ∪H (cid:16) ( b P i ) − , ( b P i +1 ) + (cid:17) ≤
1. Let R ⊆ Cay(
G, X ∪ H ) be a geodesicwith R − = ( b P i ) − and R + = ( b P i +1 ) + , so that | R | ≤ C := b P i b P i +1 R is a closed path inCay( G, X ∪ H ) of length ≤ P ′ i , P ′ i +1 and R ′ in Γ( N ) such that P ′ i P ′ i +1 R ′ is a closed path in Γ( N )that has no parabolic shortenings whose derived path is C , as follows (see Figure 6(a)). For i ′ ∈ { i, i + 1 } we take P ′ i ′ = P i ′ if P i ′ is not an extended subpath of P . Otherwise, P i ′ is anextended subpath of P of the form e T ′ e , where e and e are connecting edges and T ′ iscontained in a copy Γ ⊆ Γ( N ) of Γ j,N for some j ∈ { , . . . , m } ; we then take P ′ i ′ = e T e ,where T is a geodesic in Γ with T − = T ′− and T + = T ′ + (see Figure 5(a) for the case i ′ = 1). We take R ′ to be a path such that c R ′ = R , so that R ′ is a trivial path if | R | = 0, and R ′ consistsof two free edges if R is labelled by an element of X ; otherwise (if R is an e H j -edge) R ′ = eSe ′ for some connecting edges e , e ′ and a path S ⊆ Γ for some copy Γ of Γ j,N in Γ( N ) such that S is geodesic in Γ .In order to obtain a contradiction, we now study the components of C . If all components of C are isolated and of length : In this case, the subpath R is eithertrivial (in which case | R ′ | = 0), or an edge labelled by an element of X (in which case | R ′ | = 2), or an isolated component of C (in which case, by Lemma 3.3 and since thepath P ′ i P ′ i +1 R ′ has no parabolic shortenings, | R ′ | ≤ | R ′ | ≤ | b P ′ i | , | b P ′ i +1 | ≤
3. It then follows from Lemma 3.4 that neither P i nor P i +1 canbe an extended subpath of P : thus, it follows from our construction that P ′ i = P i and P ′ i +1 = P i +1 .We therefore have | P i | , | P i +1 | , | R ′ | ≤
3. But since | b P i | = 1, we have ( P i ) − = ( P i ) + ,and as both of these vertices are free and no two free vertices of Γ( N ) are adjacent, itfollows that | P i | ≥
2; similarly, | P i +1 | ≥
2. We thus have | R ′ | ≤ < ≤ | P i P i +1 | , andeither | P i P i +1 | ≤ | P i P i +1 | = 6 ≥ | R ′ | + 3. As P i P i +1 is a subpath of P and as R ′± = ( P i P i +1 ) ± , this contradicts the fact that P is a 5-local geodesic. If b P i and b P i +1 are in the same component of C : In this case we have P ′ i = e T e and P ′ i +1 = e T e where e , e and e are connecting edges, and T , T ⊆ Γ for somecopy Γ of Γ j,N in Γ( N ): see Figure 6(b). By construction, we then have P i = e T ′ e and P i +1 = e T ′ e for some paths T ′ , T ′ ⊆ Γ , and so e e is a subpath of P . Thiscontradicts the fact that P is a 2-local geodesic. Otherwise:
It follows that either b P i and R are connected components, or b P i +1 and R belong tothe same component of C – but not both. Without loss of generality, suppose that thelatter is true. Then R ′ = e T e and P ′ i +1 = e T e where e , e and e are connectingedges, and T , T ⊆ Γ for some copy Γ of Γ j,N in Γ( N ): see Figure 6(c). Moreover, P i is either an edge labelled by an element of X (in which case | P ′ i | = 2) or an isolatedcomponent of C (in which case, by Lemma 3.3, | P ′ i | ≤
3) – therefore, | P ′ i | ≤ P i cannot be an extended subpath of P , and so P ′ i = P i .Now let T ⊆ Γ be a geodesic in Γ with T − = ( T ) − and T + = ( T ) − , and let R ′′ = e T e . Then C ′ := P i R ′′ is a closed path in Γ( N ) with no parabolic shortenings,and c R ′′ is an isolated component of c C ′ : therefore, by Lemma 3.3, we have | R ′′ | ≤ | T | ≤
1. Now consider the paths e T and P i e : we have ( e T ) ± = ( P i e ) ± ,and, by construction, P i e is a subpath of P . We have | P i e | = | P i | + 1 ≤
4; on theother hand, as ( P i ) − , ( P i ) + are distinct free vertices of Γ( N ), we have | P i | ≥ | e T | = 2 < ≤ | P i e | . This contradicts the fact that P is a 4-local geodesic. (cid:3) Finally, we show that a word that belongs to the set b Φ defined in Proposition 3.1 cannotlabel the derived path of a geodesic in Γ( N ). Lemma 3.6.
Let P ⊆ Γ( N ) be a path such that b P is labelled by a word in b Φ . Then P is not ageodesic. P ′ P P n − P ′′ Z P − W ′ U U k W ′′ Figure 7.
An example of the paths U (blue) and P (red) in the proof ofLemma 3.6. In this case, P is an extended subpath of P , and P n is trivial. ELLY GROUPS, COARSE HELLY GROUPS, AND RELATIVE HYPERBOLICITY 15
Proof.
Note that, as b Φ only consists of non-geodesic words, we have | b P | ≥
2, and so P containsat least one free vertex. Let P = P ′ P · · · P n − P ′′ , P and P n be as in Definition 2.14, andlet P ∈ b Φ be the word labelling b P . Let U P be the word chosen in Assumption 3.2(ii), and let U P ⊆ Cay(
G, X ∪ H ) be a path labelled by U P with ( U P ) − = b P − and ( U P ) + = b P + .Now write U P = e · · · e k , where each e i is an edge in Cay( G, X ∪ H ). We define a path U ⊆ Γ( N ) as U = W ′ U · · · U k W ′′ , where (see Figure 7):(i) W ′ = Z P − if P is an extended subpath of P , and W ′ = P ′ otherwise (i.e. if P is trivial);(ii) for 1 ≤ i ≤ k , U i is a path with no parabolic shortenings and with ( U i ) − and ( U i ) + freesuch that b U i = e i : that is, U i consists of two free edges if e i is labelled by some x ∈ X ,and all non-endpoint vertices of U i are H j -internal if e i is labelled by some h ∈ e H j ;(iii) W ′′ = Z P + if P n is an extended subpath of P , and W ′′ = P ′′ otherwise.It follows from the construction that U ± = P ± and that b U = U P . We aim to show that | U | < | P | .Note first that for 1 ≤ i ≤ k , the edge e i is labelled either by an element of X (in whichcase | U i | = 2) or by some element h ∈ e H j (in which case by Assumption 3.2(ii) we have d Γ j ( v j , v j · h ) ≤ N and hence, as U i has no parabolic shortenings, | U i | ≤ | U i | ≤ i . We thus have(2) | U | = | W ′ | + | W ′′ | + k X i =1 | U i | ≤ | W ′ | + | W ′′ | + 3 k. On the other hand, the path P is either an extended subpath of P (in which case, by theconstruction in Definition 2.14, we have | P ′ | ≥ | b P | = 1, and | W ′ | = | Z P − | ≤
2: the latterfollows from Assumption 3.2(i)), or trivial (in which case we have | P ′ | = | W ′ | and | b P | = 0).Therefore, we have | P ′ | ≥ | W ′ | + 2 | b P | in either case; similarly, we have | P ′′ | ≥ | W ′′ | + 2 | b P n | .Moreover, since by construction we have | P i | ≥ ≤ i ≤ n − P i ) − and ( P i ) + are free, we actually have | P i | ≥ ≤ i ≤ n −
1. Therefore,(3) | P | = | P ′ | + | P ′′ | + n − X i =2 | P i | ≥ (cid:16) | W ′ | + 2 | b P | (cid:17) + (cid:16) | W ′′ | + 2 | b P n | (cid:17) + 2( n − | W ′ | + | W ′′ | + 2 (cid:16) | b P | + ( n −
2) + | b P n | (cid:17) ≥ | W ′ | + | W ′′ | + 2 | b P | , where the last inequality follows since | b P i | ≤ ≤ i ≤ n − b P is labelled by an element of b Φ, it follows from Proposition 3.1 that2 | b P | > d X ∪H ( b P − , b P + ) = 3 | U P | = 3 k . This implies, together with (2) and (3), that | P | ≥ | W ′ | + | W ′′ | + 2 | b P | > | W ′ | + | W ′′ | + 3 k ≥ | U | . Thus P is not a geodesic, as required. (cid:3) Finally, we use Propositions 3.1 and 3.5 together with Lemma 3.6 to prove Theorem 1.5.
Proof of Theorem 1.5.
Let λ ≥ c ≥ b Φ be the constants and the set of words over X ∪ H given in Proposition 3.1, and let N ≥ = A ∪ B , where(i) A is the set of paths P ⊆ Γ( N ) with no parabolic shortenings such that b P is labelledby a word in b Φ, and(ii) B is the set of non-geodesic paths P ⊆ Γ( N ) of length | P | ≤ A and B are G -invariant. Moreover, note that, by Lemma 2.13, thegraph Γ( N ) is proper (as a metric space) and the action G y Γ( N ) is geometric. Now it is easyto see from Definition 2.14 that given a word Q over X ∪ H there is a bound on the lengths ofpaths Q ⊆ Γ( N ) such that b Q is labelled by Q . As b Φ is finite, this implies that A is a union offinitely many G -orbits. Similarly, we can see that B consists of finitely many G -orbits, and so the orbit space Φ /G is finite. We set Φ to be a (finite) set of paths in Γ( N ) consisting of onerepresentative for each G -orbit in Φ .Now let P ⊆ Γ( N ) be a path in Γ( N ) that has no parabolic shortenings and does not containany G -translate of a path in Φ (i.e. any path in Φ ) as a subpath. It then follows that P is5-local geodesic, and that b P does not have a subpath labelled by an element of b Φ (as, if it did,any such subpath could be expressed as b Q for a subpath Q ⊆ P ). By Proposition 3.5, b P is a2-local geodesic, and thus, by Proposition 3.1, it is a ( λ, c )-quasi-geodesic that does not vertexbacktrack (and so does not backtrack). This establishes the result. (cid:3) Hellyness
In this section, we prove Proposition 4.5, implying immediately Theorem 1.2 and, conse-quently, we prove Theorem 1.1. We first recall a few definitions, following [CCG + Definition 4.1.
Let Γ be a graph.(i) Given β ≥
1, we say Γ has β -stable intervals (or simply stable intervals ) if for anygeodesic path P ⊆ Γ, any vertex w ∈ P and any u ∈ V (Γ) with d Γ ( u, P − ) = 1,there exists a geodesic Q ⊆ Γ and a vertex v ∈ Q such that Q − = u , Q + = P + and d Γ ( w, v ) ≤ β .(ii) Given ρ ≥ w ∈ V (Γ), the ball B ρ ( w ) = B ρ ( w ; Γ) is the set of all u ∈ V (Γ) suchthat d Γ ( u, w ) ≤ ρ . Given a constant ξ ≥ B = { B ρ i ( w i ) | i ∈ I} ofballs in Γ, we say B satisfies the ξ -coarse Helly property if T i ∈I B ρ i + ξ ( w i ) = ∅ . Wesay Γ is coarse Helly (respectively, Helly ) if there exists a constant ξ ≥ ξ -coarse Helly property(respectively, the 0-coarse Helly property).(iii) A group G is said to be Helly (respectively, coarse Helly ) if it acts geometrically , thatis, properly discontinuously and cocompactly, by graph automorphisms on a Helly (re-spectively, coarse Helly) graph.The merit of these definitions is supported by the following result.
Theorem 4.2 ([CCG +
20, Theorem 1.2]) . If G is a group acting geometrically on a coarse Hellygraph that has stable intervals, then G is Helly. We now start our proof of Theorem 1.1. Throughout the remainder of this section, we adoptthe following terminology.Let G be a finitely generated group (with a finite generating set X , say), and suppose that G is hyperbolic relative to a collection of its subgroups. As G is finitely generated, such acollection is finite (see [Osi06, Corollary 2.48]): H , . . . , H m , say. Let e H , . . . , e H m be isomorphiccopies of H , . . . , H m (respectively), ǫ : F ( X ) ∗ ( ∗ mj =1 e H j ) → G the canonical surjection, and H = F mj =1 ( e H j \ { } ), as in Section 2.2.Furthermore, let Γ , . . . , Γ m be proper graphs such that e H j acts on Γ j geometrically (for each j ); let v j ∈ V (Γ j ) be (fixed, but arbitrarily chosen) basepoints. We fix constants λ, N ≥ c ≥ j,N (respectively, Γ( N )) with geometricactions of e H j (respectively, G ) as in Section 2.3.The idea of our proof of Theorem 1.1 is to first use Theorem 1.5 to transform geodesics inΓ( N ) into ( λ, c )-quasi-geodesics in Cay( G, X ∪H ), and then use Theorem 2.6 or Proposition 2.10to show that if each Γ j is coarse Helly (respectively, has stable intervals), then Γ( N ) is coarseHelly (respectively, has stable intervals) as well. Theorem 1.1 will then follow from Theorem 4.2.4.1. Stable intervals.
We first show how we can pass the stable interval property from Γ j toΓ j,N , as follows. Lemma 4.3.
Let β ≥ . If Γ j has β -stable intervals, then Γ j,N has (3 β + 1) -stable intervals.Proof. We first prove the following Claim.
ELLY GROUPS, COARSE HELLY GROUPS, AND RELATIVE HYPERBOLICITY 17 QR ′ R e Swv ′ v ( ∈ R ) v ( ∈ S ) (a) The proof of the claim. Q ′ w Q ′ ℓ Q ′ k P ′ v P v P n u w ′ v n (b) The paths R (red) and Q ′ (blue) in Γ j . Figure 8.
The proof of Lemma 4.3.
Claim.
Let k ≥ . If P ⊆ Γ j is a path of length d Γ j ( P − , P + ) + k and v ∈ P is a vertex, thenthere exists a geodesic Q ⊆ Γ j with Q ± = P ± and a vertex w ∈ Q such that d Γ j ( v, w ) ≤ βk .Proof of Claim. By induction on k . The base case, k = 0, is trivial.Suppose that the claim holds for k ≤ r −
1, for some r ≥
1. Let P ⊆ Γ j be a path of length d Γ j ( P − , P + ) + r , and let v ∈ P be a vertex. Let R ⊆ P be the longest geodesic subpath of P with R − = P − . Thus Re ⊆ P is a non-geodesic subpath for an edge e ⊆ P . Let S ⊆ P be thesubpath such that P = ReS ; see Figure 8(a).Suppose first that v / ∈ R . Then we have v ∈ S , and so v ∈ R ′ S , where R ′ ⊆ Γ j is any geodesicwith R ′± = ( Re ) ± . Moreover, we have(4) | R ′ S | = | R ′ | + | S | ≤ | Re | − | S | = | P | − d Γ j (( R ′ S ) − , ( R ′ S ) + ) + ( r − . Thus, by the inductive hypothesis, there exists a geodesic Q ⊆ Γ j with Q ± = ( R ′ S ) ± = P ± anda vertex w ∈ Q with d Γ j ( v, w ) ≤ β ( r − < βr , as claimed.Suppose now that v ∈ R . Since Γ j has β -stable intervals, there exists a geodesic R ′ ⊆ Γ j with R ′± = ( Re ) ± and a vertex v ′ ⊆ R ′ such that d Γ j ( v, v ′ ) ≤ β . By (4), we can apply theinductive hypothesis to R ′ S ; in particular, since v ′ ∈ R ′ S , there exists a geodesic Q ⊆ Γ j with Q ± = ( R ′ S ) ± = P ± and a vertex w ∈ Q with d Γ j ( v ′ , w ) ≤ β ( r − d Γ j ( v, w ) ≤ d Γ j ( v, v ′ ) + d Γ j ( v ′ , w ) ≤ βr , as claimed. (cid:4) Now let P ⊆ Γ j,N be a geodesic, let u ∈ V (Γ j,N ) be a vertex with d Γ j,N ( u, P − ) = 1, and let v ∈ P be any vertex. Let n = | P | and let P − = v , . . . , v n = P + be the vertices of P , so that d Γ j,N ( v i − , v i ) = 1 for each i . By construction, it then follows that u, v , . . . , v n are in V (Γ j ),and that there exist paths P ′ , P , . . . , P n ⊆ Γ j such that we have ( P i ) − = v i − , ( P i ) + = v i and | P i | ≤ N for each i , as well as P ′− = u , P ′ + = v and | P ′ | ≤ N .Consider the path R = P ′ P · · · P n ⊆ Γ j . By construction, v ∈ R , and | R | = | P ′ | + P ni =1 | P i | ≤ ( n + 1) N . On the other hand, since P ⊆ Γ j,N is a geodesic, we have n = | P | = d Γ j,N ( v , v n ) = (cid:6) d Γ j ( v , v n ) /N (cid:7) ; therefore, d Γ j ( v , v n ) > ( n − N . It thus follows that d Γ j ( R − , R + ) ≥ d Γ j ( v , v n ) − d Γ j ( u, v ) > ( n − N − N = ( n − N ≥ | R | − N. Therefore, by the Claim, there exists a geodesic Q ′ ⊆ Γ j with Q ′± = R ± and a vertex w ′ ∈ Q ′ such that d Γ j ( v, w ′ ) ≤ βN ; see Figure 8(b).Finally, we can write Q ′ = Q ′ · · · Q ′ k , where k = ⌈| Q ′ | /N ⌉ , with | Q ′ i | ≤ N for each i . Byconstruction, there exist edges e , . . . , e k ⊆ Γ j,N with ( e i ) ± = ( Q ′ i ) ± for each i . We have | e · · · e k | = k = (cid:6) d Γ j ( Q ′− , Q ′ + ) /N (cid:7) since Q ′ ⊆ Γ j is a geodesic, and so Q := e · · · e k ⊆ Γ j,N is ageodesic. Moreover, since w ′ ∈ Q ′ we have w ′ ∈ Q ′ ℓ for some ℓ , and so d Γ j ( w ′ , w ) ≤ | Q ′ ℓ | ≤ N ,where w = ( Q ′ ℓ ) − ; note that w is a vertex of Q . Thus Q is a geodesic in Γ j,N with Q − = u and Q + = P + , and w ∈ Q is a vertex such that d Γ j ( v, w ) ≤ d Γ j ( v, w ′ ) + d Γ j ( w ′ , w ) ≤ βN + N, and so d Γ j,N ( v, w ) = (cid:6) d Γ j ( v, w ) /N (cid:7) ≤ β +1. This proves that Γ j,N has (3 β +1)-stable intervals,as required. (cid:3) Proposition 4.4.
If each Γ j has stable intervals, then so does Γ( N ) . R − = [( P − , x )] P − = b P − ǫ ( x ) P − b R − Γ e P e R R − P − = b P − b R − Γ e P e R P − R − b P − b R − (a) Bounding d X ( b P − , b R − ): P − free, R − medial (left); P − free, R − ∈ Γ (centre); P − , R − ∈ Γ (right). Γ e e ′ f f ′ e P ′′ i w e ′ R ′ i ′ g hgh g h g (b) The paths P i and R i ′ . P ′′ i wS ′ vST T ′ (c) ‘Moving’ along T and T ′ . Figure 9.
The proof of Proposition 4.4. Colours are the same as in Figure 2.
Proof.
Since each Γ j,N is locally finite and the action e H j y Γ j,N is properly discontinuous, thereare finitely many elements h ∈ e H j satisfying d Γ j,N ( v j , v j · h ) ≤
5. We may therefore choose aconstant k ≥ d X (1 , h ) ≤ k whenever j ∈ { , . . . , m } and h ∈ e H j are such that d Γ j,N ( v j , v j · h ) ≤
5. Let ε = ε ( λ, c, k ) ≥ X is finite and, for each j , the homomorphism ǫ | e H j is injective, there are finitely manyelements h ∈ e H j such that d X (1 , ǫ ( h )) ≤ ε . We may thus choose a constant ε ≥ d Γ j,N ( v j , v j · h ) ≤ ε whenever j ∈ { , . . . , m } and h ∈ e H j are such that d X (1 , ǫ ( h )) ≤ ε .Let β ≥ , . . . , Γ m all have β -stable intervals. We set β := max (cid:26) ε ε + 5 , (3 β + 1)(2 ε + 4) (cid:27) . We aim to show that Γ( N ) has β -stable intervals. In particular, let P ⊆ Γ( N ) be a geodesic,let w ∈ P be a vertex, and let w ′ ∈ V (Γ( N )) be such that d Γ( N ) ( P − , w ′ ) = 1. We will find ageodesic Q ⊆ Γ( N ) with Q − = w ′ and Q + = P + and a vertex v ∈ Q such that d Γ( N ) ( w, v ) ≤ β .Let P = P ′ P · · · P n − P ′′ and P , P n ⊆ Γ( N ) be as in Definition 2.14. Let R ⊆ Γ( N )be an arbitrary geodesic with R − = u and R + = P + , and, similarly to the case of P , let R = R ′ R · · · R ℓ − R ′′ and R , R ℓ ⊆ Γ( N ) be as in Definition 2.14.Note first that since P and R are geodesics, by Theorem 1.5 the derived paths b P and b R are2-local geodesic ( λ, c )-quasi-geodesics that do not backtrack. We furthermore claim that b P and b R are k -similar. Indeed, we have d X ( b P − , b R − ) ≤ k (see Figure 9(a)):(i) If either P − or R − is a medial vertex, then the other one is free, and d X ( b P − , b R − ) ≤ P − and R − is H j -internal (for some j ) – and so belongs to somecopy Γ of Γ j,N in Γ( N ) – and the other one is either in Γ as well or free. Then, byconstruction, there exist connecting edges e P and e R in Γ( N ) such that ( e P ) − = b P − ,( e R ) − = b R − and ( e P ) + , ( e R ) + ∈ Γ . Moreover, if P − is H j -internal, then by constructionthere exists a path Q ⊆ Γ with Q − = P − , Q + = ( e P ) + , and | Q | ≤
2, whereas if P − ELLY GROUPS, COARSE HELLY GROUPS, AND RELATIVE HYPERBOLICITY 19 is free then ( e P ) + = R − ; similarly for R − . It follows that d Γ (( e P ) + , ( e R ) + ) ≤ d X ( b P − , b R − ) ≤ k , as required.A similar argument shows that d X ( P + , R + ) ≤ k . We may thus apply Theorem 2.6 to the paths P and R .We argue in two parts, based on the minimal distance between w and a vertex of b P . If d Γ( N ) ( w, w ′ ) ≤ ε + 3 for some vertex w ′ of b P : Since b P is a 2-local geodesic, all vertices of b P are phase – in particular, w ′ ∈ b P is phase. It then follows by Theorem 2.6(i) that d X ( w ′ , v ′ ) ≤ ε for some phase vertex v ′ of b R . In particular, there is a path in Γ( N )from w ′ to v ′ consisting of ≤ ε free edges, and so we have d Γ( N ) ( w ′ , v ′ ) ≤ ε . Byconstruction, every vertex of b R is distance ≤ N )) from a vertex of R , andso d Γ( N ) ( v ′ , v ) ≤ v ∈ R . Therefore, d Γ( N ) ( w, v ) ≤ d Γ( N ) ( w, w ′ ) + d Γ( N ) ( w ′ , v ′ ) + d Γ( N ) ( v ′ , v ) ≤ (cid:18) ε (cid:19) + 2 ε + 2 ≤ β, as required. Otherwise:
Note that, by construction, any vertex of P that is not a vertex of P · · · P n mustbe distance ≤ b P − or b P + . Since d Γ( N ) ( w, b P ± ) > ε + 3 ≥
3, itfollows that w is a vertex of P · · · P n , and so w ∈ P i for some i . By our construction(see Definition 2.14), ( P i ) − and ( P i ) + are vertices of b P , and either P i is a path oflength ≤ P i are H j -internal for some (fixed) j . But | P i | ≥ d Γ( N ) (( P i ) − , w )+ d Γ( N ) ( w, ( P i ) + ) > ε +3) >
4, and so we must have P i = eP ′ i e ′ ,where e, e ′ are connecting edges and P ′ i is a path with | P ′ i | > g -copy Γ ofΓ j,N for some g ∈ G . In particular, without loss of generality we have e = { g, ( H j g, u ) } and e ′ = { hg, ( H j g, u ′ ) } for some h ∈ H j and u, u ′ ∈ V (Γ j ).We now claim that d Γ( N ) (( P ′ i ) − , ( P ′ i ) + ) > ε . Indeed, let e and e ′ be the first andthe last edges of P ′ i , so that P ′ i = e P ′′ i e ′ for some path P ′′ i ⊆ Γ ; see Figure 9(b). ByDefinition 2.14, P ′′ i is a subpath of P and so a geodesic in Γ( N ). Moreover, since d Γ( N ) ( w, ( P i ) ± ) > ε + 3 > w = ( P ′ i ) ± , and so w ∈ P ′′ i ; furthermore, d Γ( N ) ( w, ( P ′′ i ) ± ) > ε + 1, implying that d Γ( N ) (cid:0) ( P ′ i ) − , ( P ′ i ) + (cid:1) ≥ d Γ( N ) (cid:0) ( P ′′ i ) − , ( P ′′ i ) + (cid:1) − d Γ( N ) (cid:0) ( P ′ i ) − , ( P ′′ i ) − (cid:1) − d Γ( N ) (cid:0) ( P ′ i ) + , ( P ′′ i ) + (cid:1) = | P ′′ i | − d Γ( N ) (cid:0) ( P ′′ i ) − , w (cid:1) + d Γ( N ) (cid:0) w, ( P ′′ i ) + (cid:1) − > (cid:18) ε (cid:19) − ε, as claimed.Since ( P ′ i ) − = ( H j g, u ) and ( P ′ i ) + = ( H j g, u ′ ), it follows that d Γ j,N ( u, u ′ ) > ε . There-fore, by the choice of ε , it follows that d X (cid:16) ( b P i ) − , ( b P i ) + (cid:17) = d X ( g, hg ) = d X (1 , h ) > ε .Therefore, by Theorem 2.6(ii), b P i is connected to a component of b R , that is, to b R i ′ forsome i ′ ∈ { , . . . , ℓ } . Furthermore, Theorem 2.6(iii) implies that d X (cid:16) ( b P i ) − , ( b R i ′ ) − (cid:17) ≤ ε and d X (cid:16) ( b P i ) + , ( b R i ′ ) + (cid:17) ≤ ε . By the construction (Definition 2.14), we have R i ′ = f R ′ i ′ f ′ , where f = { h g, ( H j g, t ) } and f ′ = { h g, ( H j g, t ′ ) } are connecting edges (here h , h ∈ H j and t, t ′ ∈ V (Γ j )), and R ′ i ′ ⊆ Γ ; see Figure 9(b). It then follows that d X (1 , h ) = d X ( g, h g ) ≤ ε and so, by the choice of ε , we have d Γ j,N ( u, t ) ≤ ε ; conse-quently, d Γ (cid:0) ( P ′ i ) − , ( R ′ i ′ ) − (cid:1) = d Γ j,N ( u, t ) ≤ ε . Similarly, we have d Γ (cid:0) ( P ′ i ) + , ( R ′ i ′ ) + (cid:1) ≤ ε . Now by the construction, as the vertex ( R ′ i ′ ) − belongs to Γ , it is distance ≤ s ∈ Γ ∩ R ; similarly, d Γ (cid:0) ( R ′ i ′ ) + , s ′ (cid:1) ≤ s ′ ∈ Γ ∩ R .Furthermore, there exists subpath S of R such that S ⊆ Γ , S − = s and S + = s ′ . Note that we have d Γ (cid:0) ( P ′′ i ) − , S − (cid:1) ≤ d Γ (cid:0) ( P ′′ i ) − , ( P ′ i ) − (cid:1) + d Γ (cid:0) ( P ′ i ) − , ( R ′ i ′ ) − (cid:1) + d Γ (cid:0) ( R ′ i ′ ) − , S − (cid:1) ≤ ε + 1 = ε + 2 , and similarly d Γ (( P ′′ i ) + , S + ) ≤ ε + 2. Hence there exist paths T, T ′ ⊆ Γ such that T − = ( P ′′ i ) − , T + = S − , T ′− = ( P ′′ i ) + , T ′ + = S + , and | T | , | T ′ | ≤ ε + 2.Finally, note that the graph Γ is isomorphic to Γ j,N . By Lemma 4.3 and the choiceof β , it follows that Γ has (3 β + 1)-stable intervals. Since P ′′ i ⊆ Γ is a geodesic and w ∈ P ′′ i , we may use this fact | T | + | T ′ | times, ‘moving’ the endpoints of a geodesicjoining a vertex of T to a vertex of T ′ along the paths T and T ′ (see Figure 9(c)), tofind a geodesic S ′ ⊆ Γ and a vertex v ∈ S ′ such that S ′− = S − , S ′ + = S + and d Γ ( w, v ) ≤ (3 β + 1)( | T | + | T ′ | ) ≤ (3 β + 1)(2 ε + 4) ≤ β. Since S ⊆ Γ is a geodesic in Γ( N ) (and so in Γ ), it follows that | S ′ | = | S | . If we expressthe geodesic R ⊆ Γ( N ) as R = S SS for some paths S , S ⊆ Γ( N ), it then followsthat the path Q = S S ′ S ⊆ Γ( N ) is a geodesic as well, and we have Q − = R − = w ′ and Q + = R + = P + . Since v ∈ Q , this establishes the result. (cid:3) The coarse Helly property.
The remainder of this section is dedicated to a proof of thefollowing result which implies immediately Theorem 1.2. Using it, at the end of the subsectionwe prove Theorem 1.1.
Proposition 4.5.
If each Γ j is coarse Helly, then so is Γ( N ) . We start by making the following elementary observation.
Lemma 4.6.
Let ξ ≥ . If Γ j is ξ -coarse Helly, then Γ j,N is ⌈ ξ/N ⌉ -coarse Helly.Proof. Note that B ρ ( w ; Γ j,N ) = B ρN ( w ; Γ j ) for each w ∈ V (Γ j ) and ρ ≥
0. Thus, forany collection { B ρ i ( w i ; Γ j,N ) | i ∈ I} of pairwise intersecting balls in Γ j,N , the collection { B ρ i N ( w i ; Γ j ) | i ∈ I} is a collection of pairwise intersecting balls in Γ j . By the ξ -coarse Hellyproperty, it follows that there exists v ∈ V (Γ j ) with d Γ j ( v, w i ) ≤ ρ i N + ξ for each i ∈ I . Wethus have d Γ j,N ( v, w i ) ≤ ⌈ ( ρ i N + ξ ) /N ⌉ = ρ i + ⌈ ξ/N ⌉ for each i ∈ I , proving the ⌈ ξ/N ⌉ -coarseHelly property for the collection of balls in Γ j,N . (cid:3) Now as before, let λ ≥ c ≥ P ⊆ Γ( N ),the path b P ⊆ Cay(
G, X ∪ H ) is a 2-local geodesic ( λ, c )-quasi-geodesic that does not backtrack:such constants exist by Theorem 1.5. Let µ = µ ( λ, c ) be the constant given by Proposition 2.10.Since X is finite and ǫ | e H j is injective for each j , there are finitely many elements h ∈ H suchthat | ǫ ( h ) | X ≤ µ . We may therefore choose a constant µ ≥ d Γ j,N ( v j , v j · h ) ≤ µ whenever h ∈ e H j satisfies | ǫ ( h ) | X ≤ µ .For the remainder of this section, we assume that Γ , . . . , Γ m are coarse Helly. Let ξ ≥ , . . . , Γ m are all ξ -coarse Helly, and set(5) ξ = max (cid:26) µ µ + 5 , µ (cid:24) ξ N (cid:25) + 4 (cid:27) . We aim to show that every collection of pairwise intersecting balls in Γ( N ) satisfies the ξ -coarseHelly property. This will establish Proposition 4.5.Thus, let B ′ = n B ρ ′ i ( x ′ i ; Γ( N )) (cid:12)(cid:12)(cid:12) i ∈ I o be a collection of pairwise intersecting balls in Γ( N ).By the choice of N (see Assumption 3.2(i)), any vertex of Γ( N ) is distance ≤ i ∈ I we may pick a free vertex x i ∈ V (Γ( N )) such that d Γ( N ) ( x i , x ′ i ) ≤ ρ i = ρ ′ i + 2 (for all i ∈ I ), we see that the collection B = { B ρ i ( x i ; Γ( N )) | i ∈ I} ofballs in Γ( N ) satisfies B ρ i ( x i ) ⊇ B ρ ′ i ( x ′ i ) for all i ∈ I , and hence, as the balls in B ′ have pairwisenon-empty intersections, so do the balls in B . Moreover, since B ρ i + ξ − ( x i ) = B ρ ′ i + ξ − ( x i ) ⊆ B ρ ′ i + ξ ( x ′ i ) for each i , in order to show that B ′ satisfies the ξ -coarse Helly property, it is enough ELLY GROUPS, COARSE HELLY GROUPS, AND RELATIVE HYPERBOLICITY 21 to show that B satisfies the ( ξ − y ∈ V (Γ( N )) be an arbitrary free vertex ( y = 1 G , say), and consider the set D = { d Γ( N ) ( y, x i ) − ρ i | i ∈ I} ⊆ Z . Note that D is bounded from above: indeed, for any (fixed) j ∈ I and any i ∈ I we have d Γ( N ) ( y, x i ) ≤ d Γ( N ) ( y, x j ) + d Γ( N ) ( x j , x i ) ≤ d Γ( N ) ( y, x j ) + ρ j + ρ i since B ρ j ( x j ) ∩ B ρ i ( x i ) = ∅ , and hence D ≤ d Γ( N ) ( y, x j ) + ρ j for any D ∈ D . We may thereforefix an index I ∈ I such that d Γ( N ) ( y, x I ) − ρ I = max D .We pick a geodesic R ⊆ Γ( N ) and, for each i ∈ I , geodesics P i , Q i ⊆ Γ( N ), such that( P i ) + = ( Q i ) − = x i , ( Q i ) + = R − = y and R + = ( P i ) − = x I , so that P i Q i R is a geodesic trianglein Γ( N ) with vertices x I , x i and y ; note that we do not exclude the ‘degenerate’ cases when x i ∈ { y, x I } . If | R | < ρ I , it then follows that D < D ∈ D and so y ∈ T B ∈B B , provingthe Helly property (and so the ( ξ − B . We may therefore withoutloss of generality assume that | R | ≥ ρ I . Let z ∈ R be the vertex such that d Γ( N ) ( x I , z ) = ρ I ;see Figure 10. R Q i P i ρ I yzx I x i z ′ v i ( ∈ P i ) v i ( ∈ Q i ) Figure 10.
The proof of Proposition 4.5: general setup and Lemma 4.7.Our proof of the ( ξ − B splits into two cases, based on whetheror not z is ‘close’ to a vertex of b R . Lemma 4.7. If d Γ( N ) ( z, z ′ ) ≤ µ + 1 for some vertex z ′ ∈ b R , then B satisfies the (cid:16) µ + 4 µ + 1 (cid:17) -coarse Helly property.Proof. Fix i ∈ I , and let P = P i and Q = Q i . By Theorem 1.5, since P, Q, R ⊆ Γ( N )are geodesics, b P , b Q, b R ⊆ Cay(
G, X ∪ H ) are 2-local geodesic ( λ, c )-quasi-geodesics that do notbacktrack; in particular, every vertex of b R is phase. Moreover, as the endpoints of P , Q and R are free vertices, we have b P ± = P ± , b Q ± = Q ± and b R ± = R ± , and so b P b Q b R is a triangle inCay( G, X ∪ H ).It then follows from Proposition 2.10(i) that there exists a vertex v = v i of b P or of b Q such that d X ( z ′ , v ) ≤ µ . Since vertices of b P (respectively, b Q ) are precisely the free verticesof P (respectively, Q ), it follows that either v ∈ P or v ∈ Q ; see Figure 10. We claim that d Γ( N ) ( x i , v ) ≤ ρ i + d Γ( N ) ( z, v ).If v ∈ P , then note that d Γ( N ) ( x I , v ) ≥ d Γ( N ) ( x I , z ) − d Γ( N ) ( z, v ) = ρ I − d Γ( N ) ( z, v ). Since B ρ I ( x I ) ∩ B ρ i ( x i ) = ∅ , it follows that d Γ( N ) ( x i , x I ) ≤ ρ i + ρ I , and so, as P is a geodesic, d Γ( N ) ( x i , v ) = d Γ( N ) ( x i , x I ) − d Γ( N ) ( x I , v ) ≤ ( ρ i + ρ I ) − ( ρ I − d Γ( N ) ( z, v ) = ρ i + d Γ( N ) ( z, v ) , as claimed.On the other hand, if v ∈ Q , then note that d Γ( N ) ( y, z ) = d Γ( N ) ( y, x I ) − ρ I ≥ d Γ( N ) ( y, x i ) − ρ i by the choice of I . It follows that d Γ( N ) ( y, v ) ≥ d Γ( N ) ( y, z ) − d Γ( N ) ( z, v ) ≥ d Γ( N ) ( y, x i ) − ( ρ i + d Γ( N ) ( z, v )) , and so, as Q is a geodesic, we have d Γ( N ) ( x i , v ) = d Γ( N ) ( y, x i ) − d Γ( N ) ( y, v ) ≤ ρ i + d Γ( N ) ( z, v ),as claimed. Therefore, we have d Γ( N ) ( x i , v ) ≤ ρ i + d Γ( N ) ( z, v ) in either case, and hence(6) d Γ( N ) ( x i , z ′ ) ≤ d Γ( N ) ( x i , v ) + d Γ( N ) ( z ′ , v ) ≤ ρ i + d Γ( N ) ( z, v ) + d Γ( N ) ( z ′ , v ) . Now since d X ( z ′ , v ) ≤ µ , there is a path in Γ( N ) from z ′ to v consisting of ≤ µ free edges, andso d Γ( N ) ( z ′ , v ) ≤ µ . It follows that d Γ( N ) ( z, v ) ≤ d Γ( N ) ( z, z ′ ) + d Γ( N ) ( z ′ , v ) ≤ µ + 1 + 2 µ , andso (6) implies that d Γ( N ) ( x i , z ′ ) ≤ ρ i + µ + 4 µ + 1.Thus the intersection T i ∈I B ρ i + µ +4 µ +1 ( x i ; Γ( N )) contains the vertex z ′ and so is non-empty.Therefore, B satisfies the (cid:16) µ + 4 µ + 1 (cid:17) -coarse Helly property, as required. (cid:3) Lemma 4.8. If d Γ( N ) ( z, z ′ ) > µ + 1 for all z ′ ∈ b R , then B satisfies the (cid:16) µ + l ξ N m(cid:17) -coarseHelly property. R Q i P i yx I x i rr ′ p i p ′ i Γ (a) i ∈ J P \ J Q . R Q i P i yx I x i rr ′ q i q ′ i Γ (b) i ∈ J Q \ J P . R Q i P i yx I x i rr ′ p i p ′ i q i q ′ i Γ (c) i ∈ J P ∩ J Q . R Q i P i yx I x i z rr ′ p ′ i σ ′ σρ i − τ i Γ (d) Definitions of σ , σ ′ and τ i . SP i P i ′ x i x i ′ x I p i p ′ i p i ′ p ′ i ′ (e) The path S does not penetrate Γ . SP i P i ′ x i x i ′ x I p i p ′ i p i ′ p ′ i ′ s s ′ (f) The path S penetrates Γ . Figure 11.
The proof of Lemma 4.8.
Proof.
Since by construction the vertices of b R are precisely the free vertices of R , it follows thatthe vertex z ∈ R is distance > µ + 1 ≥ R . It follows that z is H j -internal (for some j ), and so z ∈ Γ , where Γ is the g -copy of Γ j,N in Γ( N ) for some g ∈ G .In particular, R penetrates Γ .Throughout the proof, we adopt the following terminology. Suppose S ⊆ Γ( N ) is a geodesicthat penetrates Γ such that S − and S + are free vertices. By Theorem 1.5, b S ⊆ Cay(
G, X ∪ H )is then a 2-local geodesic that does not backtrack. It then follows that S = S e S ′ e S , where ELLY GROUPS, COARSE HELLY GROUPS, AND RELATIVE HYPERBOLICITY 23 S ′ = S ∩ Γ , and e = { h g, ( H j g, u ) } , e = { h g, ( H j g, u ) } are connecting edges for some h , h ∈ H j and u , u ∈ V (Γ j ). In this case, we say S enters (respectively, leaves ) Γ atthe vertex S ′− (respectively, S ′ + ). The path b S ⊆ Cay(
G, X ∪ H ) then has an edge e ⊆ b S with e − = h g and e + = h g such that e is an H j -component of b S ; we say that e is the H j -componentof b S associated to Γ .The following observation follows from the choice of µ . Observation.
Let S ⊆ Γ( N ) be a geodesic that penetrates Γ with S ± free, let u and u bethe vertices at which S enters and leaves Γ , respectively, and let e be the H j -component of b S associated to Γ . If d X ( e − , e + ) ≤ µ , then d Γ ( u , u ) ≤ µ (and therefore d Γ( N ) ( u , u ) ≤ µ ). Let r and r ′ be the vertices of Γ( N ) at which R enters and leaves Γ , respectively. Since r and r ′ are adjacent to free vertices of R , it follows from the premise that d Γ( N ) ( z, r ) , d Γ( N ) ( z, r ′ ) > µ .Therefore, as R is a geodesic, we have(7) d Γ( N ) ( r, r ′ ) = d Γ( N ) ( r, z ) + d Γ( N ) ( z, r ′ ) > µ. Now for each i ∈ I , it follows from Theorem 1.5 that b P i c Q i b R is a non-backtracking ( λ, c )-quasi-geodesic triangle in Cay( G, X ∪ H ). By (7) and the Observation, if e is the H j -componentof b R associated to Γ then d X ( e − , e + ) > µ . Therefore, it follows from Proposition 2.10(ii) that e is connected to an H j -component of either b P i or c Q i , and by construction such a componentmust be associated to Γ . In particular, either P i or Q i (or both) must penetrate Γ .Let J P ⊆ I (respectively, J Q ⊆ I ) be the set of all i ∈ I such that P i (respectively, Q i )penetrates Γ , so that I = J P ∪ J Q . For each i ∈ J P , let p i and p ′ i be the vertices of Γ( N ) atwhich P i enters and leaves Γ , respectively; similarly, for i ∈ J Q , let q i and q ′ i be the vertices ofΓ( N ) at which Q i enters and leaves Γ , respectively. By parts (iii) and (iv) of Proposition 2.10and the choice of µ (cf the Observation above), it follows that(a) If i ∈ J P \ J Q , then d Γ ( r ′ , p i ) ≤ µ and d Γ ( p ′ i , r ) ≤ µ .(b) If i ∈ J Q \ J P , then d Γ ( r ′ , q i ) ≤ µ and d Γ ( q ′ i , r ) ≤ µ .(c) If i ∈ J P ∩ J Q , then d Γ ( r ′ , p i ) ≤ µ , d Γ ( p ′ i , q i ) ≤ µ and d Γ ( q ′ i , r ) ≤ µ .This is depicted in parts (a), (b) and (c) of Figure 11.Now let σ = d Γ( N ) ( z, r ) and σ ′ = d Γ( N ) ( z, r ′ ). For each i ∈ J P , let also τ i = ρ i − d Γ( N ) ( x i , p ′ i ).Consider the following collection of balls in Γ : B = { B σ ′ ( r ′ ; Γ ) } ∪ { B τ i + µ ( p ′ i ; Γ ) | i ∈ J P } . We claim that τ i + 2 µ ≥ i ∈ J P and that the balls in B have pairwise non-emptyintersections. τ i + 2 µ ≥ for each i ∈ J P : Suppose first that i ∈ J P \ J Q ; see Figure 11(a). By the point(a) above we then have d Γ( N ) ( p ′ i , r ) ≤ µ , and so d Γ( N ) ( x I , p ′ i ) ≥ d Γ( N ) ( x I , r ) − d Γ( N ) ( p ′ i , r ) ≥ ρ I + σ − µ. Since d Γ( N ) ( x i , x I ) ≤ ρ i + ρ I and since P i is a geodesic, we thus have ρ i − τ i = d Γ( N ) ( x i , p ′ i ) = d Γ( N ) ( x i , x I ) − d Γ( N ) ( x I , p ′ i ) ≤ ( ρ i + ρ I ) − ( ρ I + σ − µ ) = ρ i − ( σ − µ ) . Therefore, τ i + 2 µ ≥ σ + µ ≥
0, as claimed.Suppose now that i ∈ J P ∩ J Q ; see Figure 11(c). It then follows from the point (c)above that d Γ( N ) ( p ′ i , q i ) ≤ µ and d Γ( N ) ( r, q ′ i ) ≤ µ . Therefore, we have(8) d Γ( N ) ( x i , q i ) ≥ d Γ( N ) ( x i , p ′ i ) − d Γ( N ) ( p ′ i , q i ) ≥ ρ i − τ i − µ and d Γ( N ) ( y, q ′ i ) ≥ d Γ( N ) ( y, x I ) − d Γ( N ) ( x I , z ) − d Γ( N ) ( z, r ) − d Γ( N ) ( r, q ′ i ) ≥ d Γ( N ) ( y, x I ) − ρ I − σ − µ ≥ d Γ( N ) ( y, x i ) − ρ i − σ − µ, where the last inequality follows from the choice of I . As Q i is a geodesic, this impliesthat d Γ( N ) ( x i , q ′ i ) = d Γ( N ) ( y, x i ) − d Γ( N ) ( y, q ′ i ) ≤ ρ i + σ + µ, and combining this with (8) gives(9) d Γ( N ) ( q i , q ′ i ) = d Γ( N ) ( x i , q ′ i ) − d Γ( N ) ( x i , q i ) ≤ ( ρ i + σ + µ ) − ( ρ i − τ i − µ ) = σ + τ i + 2 µ. Suppose for contradiction that τ i + 2 µ <
0. We then get d Γ( N ) ( x i , p ′ i ) > ρ i + 2 µ by the definition of τ i . Moreover, (9) implies that d Γ( N ) ( q i , q ′ i ) < σ , and so d Γ( N ) ( x I , p ′ i ) ≥ d Γ( N ) ( x I , r ) − d Γ( N ) ( r, q ′ i ) − d Γ( N ) ( q ′ i , q i ) − d Γ( N ) ( q i , p ′ i ) > ( ρ I + σ ) − µ − σ − µ = ρ I − µ. Therefore, as P i is a geodesic, we get d Γ( N ) ( x i , x I ) = d Γ( N ) ( x i , p ′ i ) + d Γ( N ) ( x I , p ′ i ) > ρ i + ρ I , contradicting the fact that B ρ i ( x i ) ∩ B ρ I ( x I ) = ∅ . Thus we must have τ i + 2 µ ≥
0, asclaimed. B σ ′ ( r ′ ; Γ ) ∩ B τ i + µ ( p ′ i ; Γ ) = ∅ for each i ∈ J P : As shown above, τ i + 2 µ ≥
0, and so both σ ′ and τ i + µ are non-negative. It therefore suffices to show that d Γ ( r ′ , p ′ i ) ≤ σ ′ + τ i + µ .It follows from the points (a) and (c) above that d Γ( N ) ( r ′ , p i ) ≤ d Γ ( r ′ , p i ) ≤ µ , andhence d Γ( N ) ( x I , p i ) ≥ d Γ( N ) ( x I , z ) − d Γ( N ) ( z, r ′ ) − d Γ( N ) ( r ′ , p i ) ≥ ρ I − σ ′ − µ. Since P i is a geodesic and since d Γ( N ) ( x I , x i ) ≤ ρ I + ρ i , we thus have d Γ ( p i , p ′ i ) = d Γ( N ) ( p i , p ′ i ) = d Γ( N ) ( x I , x i ) − d Γ( N ) ( x I , p i ) − d Γ( N ) ( x i , p ′ i ) ≤ ( ρ I + ρ i ) − ( ρ I − σ ′ − µ ) − ( ρ i − τ i ) = σ ′ + τ i + µ. Therefore, d Γ ( r ′ , p ′ i ) ≤ d Γ ( r ′ , p i ) + d Γ ( p i , p ′ i ) ≤ µ + ( σ ′ + τ i + µ ) ≤ σ ′ + τ i + 5 µ , as required. B τ i + µ ( p ′ i ; Γ ) ∩ B τ i ′ + µ ( p ′ i ′ ; Γ ) = ∅ for all i, i ′ ∈ J P : As shown above, τ i , τ i ′ ≥ − µ , and soboth τ i + µ and τ i ′ + µ are non-negative. It is thus enough to show that we have d Γ ( p ′ i , p ′ i ′ ) ≤ τ i + τ i ′ + 5 µ . Let S ⊆ Γ( N ) be a geodesic with S − = x i and S + = x i ′ .We will apply Proposition 2.10 to the triangle b P i b S c P i ′ ⊆ Cay(
G, X ∪ H ), which is anon-backtracking ( λ, c )-quasi-geodesic triangle by Theorem 1.5.Suppose first that S does not penetrate Γ ; see Figure 11(e). It then follows fromProposition 2.10(iii) (cf the point (a) above) that d Γ ( p ′ i , p ′ i ′ ) ≤ µ . In particular, since τ i , τ i ′ ≥ − µ we have d Γ ( p ′ i , p ′ i ′ ) ≤ µ ≤ µ + ( τ i + 2 µ ) + ( τ i ′ + 2 µ ) = τ i + τ i ′ + 5 µ, as required.Suppose now that S penetrates Γ , and let s and s ′ be the vertices at which S entersand leaves Γ , respectively; see Figure 11(f). It follows from Proposition 2.10(iv) (cf thepoint (c) above) that d Γ( N ) ( s, p ′ i ) ≤ d Γ ( s, p ′ i ) ≤ µ and d Γ( N ) ( s ′ , p ′ i ′ ) ≤ d Γ ( s ′ , p ′ i ′ ) ≤ µ .In particular, we have d Γ( N ) ( x i , s ) ≥ d Γ( N ) ( x i , p ′ i ) − d Γ( N ) ( s, p ′ i ) ≥ ρ i − τ i − µ, ELLY GROUPS, COARSE HELLY GROUPS, AND RELATIVE HYPERBOLICITY 25 and similarly d Γ( N ) ( x i ′ , s ′ ) ≥ ρ i ′ − τ i ′ − µ . Since S is a geodesic and since d Γ( N ) ( x i , x i ′ ) ≤ ρ i + ρ i ′ , it follows that d Γ ( s, s ′ ) = d Γ( N ) ( s, s ′ ) = d Γ( N ) ( x i , x i ′ ) − d Γ( N ) ( x i , s ) − d Γ( N ) ( x i ′ , s ′ ) ≤ ( ρ i + ρ i ′ ) − ( ρ i − τ i − µ ) − ( ρ i ′ − τ i ′ − µ ) = τ i + τ i ′ + 2 µ. Therefore, d Γ ( p ′ i , p ′ i ′ ) ≤ d Γ ( p ′ i , s ) + d Γ ( s, s ′ ) + d Γ ( s ′ , p ′ i ′ ) ≤ µ + ( τ i + τ i ′ + 2 µ ) + µ ≤ τ i + τ i ′ + 5 µ, as required.We have thus shown that B is a collection of pairwise intersecting balls in Γ . But sinceΓ is isomorphic to Γ j,N , it follows from Lemma 4.6 and the choice of ξ that Γ is ⌈ ξ /N ⌉ -coarse Helly. Therefore, there exists a vertex z ∈ V (Γ ) such that d Γ ( r ′ , z ) ≤ σ ′ + ⌈ ξ /N ⌉ and d Γ ( p ′ i , z ) ≤ τ i + µ + ⌈ ξ /N ⌉ for all i ∈ J P . We claim that d Γ( N ) ( x i , z ) ≤ ρ i + µ + l ξ N m for all i ∈ I : this will establish the (cid:16) µ + l ξ N m(cid:17) -coarse Helly property for B .Suppose first that i ∈ J P . We then have d Γ( N ) ( x i , z ) ≤ d Γ( N ) ( x i , p ′ i ) + d Γ( N ) ( p ′ i , z ) ≤ ( ρ i − τ i ) + (cid:18) τ i + 5 µ (cid:24) ξ N (cid:25)(cid:19) = ρ i + 5 µ (cid:24) ξ N (cid:25) , as claimed.Suppose now that i / ∈ J P ; since I = J P ∪ J Q , it follows that i ∈ J Q (see Figure 11(b)). Bythe point (b) above, we have d Γ( N ) ( r ′ , q i ) ≤ µ . Therefore, d Γ( N ) ( y, q i ) ≥ d Γ( N ) ( y, x I ) − d Γ( N ) ( x I , r ′ ) − d Γ( N ) ( r ′ , q i ) ≥ d Γ( N ) ( y, x I ) − ρ I + σ ′ − µ ≥ d Γ( N ) ( y, x i ) − ρ i + σ ′ − µ, where the last inequality follows by the choice of I . As Q i is a geodesic, this implies that d Γ( N ) ( x i , q i ) = d Γ( N ) ( y, x i ) − d Γ( N ) ( y, q i ) ≤ ρ i − σ ′ + µ, and hence d Γ( N ) ( x i , z ) ≤ d Γ( N ) ( x i , q i ) + d Γ( N ) ( q i , r ′ ) + d Γ( N ) ( r ′ , z ) ≤ ( ρ i − σ ′ + µ ) + µ + (cid:18) σ ′ + (cid:24) ξ N (cid:25)(cid:19) ≤ ρ i + 5 µ (cid:24) ξ N (cid:25) , as claimed.We thus have d Γ( N ) ( x i , z ) ≤ ρ i + µ + l ξ N m for all i ∈ I . Hence T i ∈I B ρ i + µ + l ξ N m ( x i ; Γ( N ))contains z and so is non-empty, establishing the (cid:16) µ + l ξ N m(cid:17) -coarse Helly property for B . (cid:3) Finally, we deduce the conclusions of Proposition 4.5 and Theorem 1.1.
Proof of Proposition 4.5.
By the choice of ξ in (5), it follows from Lemmas 4.7 and 4.8 that B satisfies the ( ξ − B ′ satisfies the ξ -coarse Helly property. As B ′ was an arbitrary collection of balls in Γ( N ), the conclusion follows. (cid:3) Proof of Theorem 1.1.
Suppose that the graphs Γ , . . . , Γ m as above are Helly. Then they areclearly coarse Helly; moreover, by [CCG +
20, Lemma 6.5], each Γ j has 1-stable intervals. Thus,the Theorem follows immediately from Propositions 4.4 and 4.5 together with Theorem 4.2. (cid:3) Quasiconvex subgroups of Helly groups
In this section, we prove Theorem 1.4 from the Introduction, that is, we show that if asubgroup H of a (coarse) Helly group G is, in a certain sense, quasiconvex in G , then H is(coarse) Helly. Definition 5.1.
Let Γ be a graph.(i) Given λ ≥ c ≥
0, we say a subset W ⊆ V (Γ) is ( λ, c ) -quasiconvex if there exists k = k ( λ, c ) ≥ λ, c )-quasigeodesic in Γ with endpoints in W is in the k -neighbourhood of W .(ii) Let G be a group acting on Γ geometrically. We say a subgroup H ≤ G is stronglyquasiconvex (respectively, semi-strongly quasiconvex ) with respect to Γ if some H -orbitin Γ is ( λ, c )-quasiconvex (respectively, (1 , c )-quasiconvex) for any λ ≥ c ≥ k ≥
1, weconstruct a (Vietoris-Rips) graph Γ k with V (Γ k ) = V (Γ), such that v, w ∈ V (Γ) are adjacentin Γ k if and only if d Γ ( v, w ) ≤ k . Thus Γ = Γ.Since the collection of balls in Γ k is just a subcollection of the collection of balls in Γ (for any k ≥ Lemma 5.2.
If a graph Γ is Helly, then so is Γ k for any k ≥ . (cid:3) Our proof of Theorem 1.4 is based on the following two Lemmas.
Lemma 5.3.
Let ξ ≥ , let Γ be a ξ -coarse Helly graph, and W ⊆ V (Γ) a (1 , ξ ) -quasiconvexsubset. Let k ≥ be such that every (1 , ξ ) -quasigeodesic in Γ with endpoints in W is containedin the k -neighbourhood of W . Then the full subgraph ∆ of Γ k spanned by S w ∈ W B ( w ; Γ k ) is (3 + ⌈ ξ/k ⌉ ) -coarse Helly.Proof. Let B = { B ρ i ( x i ; ∆) | i ∈ I} be a collection of pairwise intersecting balls in ∆. Bythe construction of ∆, for each i ∈ I there exists x ′ i ∈ W such that d ∆ ( x i , x ′ i ) ≤
1, and inparticular B ρ i ( x i ; ∆) ⊆ B ρ i +1 ( x ′ i ; ∆) ⊆ B ρ i +1 ( x ′ i ; Γ k ) = B ( ρ i +1) k ( x ′ i ; Γ). It follows that B ′ = { B ( ρ i +1) k ( x ′ i ; Γ) | i ∈ I} is a collection of pairwise intersecting balls.Now fix any j ∈ I , and consider D = { d Γ ( x ′ i , x ′ j ) − ( ρ i + 1) k | i ∈ I} ⊆ Z ; let δ = sup D .Since the balls in B ′ have pairwise non-empty intersections, we have δ ≤ ( ρ j + 1) k (and inparticular δ = max D ). If δ ≤
0, it then follows that d Γ ( x ′ i , x ′ j ) ≤ ( ρ i + 1) k for all i ∈ I ;therefore, since x ′ i , x ′ j ∈ W and in particular any geodesic in Γ between x ′ i and x ′ j is a path in∆, d ∆ ( x ′ i , x ′ j ) = ⌈ d Γ ( x ′ i , x ′ j ) /k ⌉ ≤ ρ i + 1 for all i ∈ I . Hence, x ′ j ∈ T i ∈I B ρ i +2 ( x i ; ∆), implyingthat B satisfies the 2-coarse Helly property. Thus, we may assume that δ > δ , we know that B ′′ = { B ( ρ i +1) k ( x ′ i ; Γ) | i ∈ I \ { j }} ∪ { B δ ( x ′ j ; Γ) } is acollection of pairwise intersecting balls. It follows by the ξ -coarse Helly property that thereexists y ∈ V (Γ) such that d Γ ( y, x ′ i ) ≤ ( ρ i + 1) k + ξ for all i ∈ I \ { j } and d Γ ( y, x ′ j ) ≤ δ + ξ .Moreover, by the choice of δ , there exists ℓ ∈ I such that d Γ ( x ′ j , x ′ ℓ ) = ( ρ ℓ +1) k + δ . Therefore, d Γ ( y, x ′ j ) + d Γ ( y, x ′ ℓ ) − d Γ ( x ′ j , x ′ ℓ ) ≤ ( δ + ξ ) + [( ρ ℓ + 1) k + ξ ] − [( ρ ℓ + 1) k + δ ] = 2 ξ, and so y lies on a (1 , ξ )-quasigeodesic in Γ from x ′ j to x ′ ℓ ; thus, y ∈ V (∆), implying that d Γ ( y, y ′ ) ≤ k for some y ′ ∈ W .By the choice of ∆, it follows that all geodesics in Γ between y ′ and x ′ i are paths in ∆,and so d ∆ ( y ′ , x ′ i ) = ⌈ d Γ ( y ′ , x ′ i ) /k ⌉ for each i ∈ I . Moreover, since δ ≤ ( ρ j + 1) k , we have d Γ ( y, x ′ i ) ≤ ( ρ i + 1) k + ξ for each i ∈ I , and so d ∆ ( y ′ , x ′ i ) ≤ (cid:24) d Γ ( y ′ , y ) + d Γ ( y, x ′ i ) k (cid:25) ≤ (cid:24) k + [( ρ i + 1) k + ξ ] k (cid:25) = ρ i + 2 + (cid:24) ξk (cid:25) for all i ∈ I . It follows that d ∆ ( y ′ , x i ) ≤ d ∆ ( y ′ , x ′ i ) + d ∆ ( x ′ i , x i ) ≤ ρ i + 3 + (cid:24) ξk (cid:25) , ELLY GROUPS, COARSE HELLY GROUPS, AND RELATIVE HYPERBOLICITY 27 which proves the (3 + ⌈ ξ/k ⌉ )-coarse Helly property for B . (cid:3) In the next Lemma, we say a graph Γ is pseudo-modular if for every triple w , w , w ∈ V (Γ)there exist geodesics P , P , P , e , , e , , e , in Γ, with | e , | = | e , | = | e , | ≤
1, such that( P i ) − = w i , ( P i ) + = ( e i,i +1 ) − = ( e i − ,i ) + , and such that P i e i,i +1 P i +1 is a geodesic in Γ for i ∈ { , , } (with indices taken modulo 3). It is well-known (see [BM86, Proposition 4])that a connected graph Γ is pseudo-modular if and only if every triple { B ′ , B ′ , B ′ } of pairwiseintersecting balls in Γ has non-empty intersection (that is, satisfies the 0-coarse Helly property).In particular, every Helly graph is pseudo-modular. Lemma 5.4.
Let Γ be a pseudo-modular graph, and W ⊆ V (Γ) a (5 , -quasiconvex subset.Let k ≥ be such that every (5 , -quasigeodesic in Γ with endpoints in W is contained inthe k -neighbourhood of W . Then the full subgraph ∆ of Γ k spanned by S w ∈ W B ( w ; Γ k ) isisometrically embedded in Γ k . P ′ v P v e e e Q P ′ w P w f f f v ′ v w ′ w Figure 12.
The proof of Lemma 5.4.
Proof.
Suppose for contradiction that ∆ is not isometrically embedded in Γ k . Thus, there exist v, w ∈ V (∆) such that d ∆ ( v, w ) > d Γ k ( v, w ); without loss of generality, assume that v and w are chosen in such a way that d Γ k ( v, w ) is as small as possible. By the definition of ∆, thereexist v ′ , w ′ ∈ W such that d Γ ( v, v ′ ) ≤ k and d Γ ( w, w ′ ) ≤ k .Since Γ is pseudo-modular, there exist geodesics P ′ v e P v , P ′ v e Q ′ and P v e Q ′ in Γ from v ′ to v , from v ′ to w and from v to w , respectively, such that | e | = | e | = | e | ≤
1. Similarly, thereexist geodesics P ′ w f P w , Qf P ′ w and Qf P w in Γ from w ′ to w , from Q ′− to w ′ and from Q ′− to w , respectively, such that | f | = | f | = | f | ≤
1: see Figure 12. Note that Q ′ is a geodesicin Γ with the same endpoints as the geodesic Qf P w : therefore, since P ′ v e Q ′ and P v e Q ′ aregeodesics in Γ, so are P ′ v e Qf P w and P v e Qf P w . In particular, the paths P ′ v e P v , P ′ w f P w , P ′ v e Q , Qf P ′ w and P v e Qf P w are all geodesics in Γ.Let p = d Γ ( v, w ) − k [ d Γ k ( v, w ) − ≤ p ≤ k . We claim that | P v e | < | P ′ v e | − k + p .Indeed, let u ∈ P v e Qf P w be the vertex with d Γ ( v, u ) = p ; it follows from the choice of p andthe fact that P v e Qf P w is a geodesic in Γ that u lies on a geodesic in Γ k between v and w .Since u is adjacent to v in Γ k and since ∆ is a full subgraph, it follows by the minimality of d Γ k ( v, w ) that u / ∈ V (∆); therefore, d Γ ( v ′ , u ) > k . In particular, since d Γ ( v ′ , v ) ≤ k and since P ′ v e P v is a geodesic in Γ, this implies that u / ∈ P v and so u ∈ Qf P w . We thus have k < d Γ ( v ′ , u ) ≤ | P ′ v e | + d Γ ( Q − , u ) = | P ′ v e | + ( p − | P v e | ) , and so | P v e | < | P ′ v e | − k + p , as claimed. Similarly, | f P w | < | f P ′ w | − k + p .We now claim that | Q | ≥ k . Indeed, since p ≤ k and | e | ≤
1, the previous paragraph impliesthat | P v e | ≤ | P ′ v | , and hence2 | P v e | ≤ | P v e | + | P ′ v | = | P v | + | e | + | P ′ v | = d Γ ( v ′ , v ) ≤ k since P ′ v e P v is a geodesic in Γ; similarly, 2 | f P w | ≤ k . Therefore, if d Γ ( v, w ) ≥ k then we have | Q | = d Γ ( v, w ) − | P v e | − | f P w | ≥ k − k − k k , as claimed. On the other hand, suppose that d Γ ( v, w ) < k . It then follows that d Γ k ( v, w ) = 2(as v and w cannot be adjacent in Γ k ), and so p < k and d Γ ( v, w ) = k + p . Then, again by theprevious paragraph, we have(10) | Q | = d Γ ( v, w ) − | P v e | − | f P w | > ( k + p ) − ( | P ′ v e | − k + p ) − ( | f P ′ w | − k + p )= 3 k − p − | P ′ v e | − | f P ′ w | . Furthermore, we have | P ′ v e | = | P ′ v | + | e | ≤ | P ′ v e P v | = d Γ ( v ′ , v ) ≤ k since P ′ v e P v is a geodesicin Γ, and similarly | f P ′ w | ≤ k . Therefore, (10) implies that | Q | > k − p . As p < k , it followsthat | Q | ≥ k , as claimed.Finally, we claim that P ′ v e Qf P ′ w is a (5 , R ⊆ P ′ v e Qf P ′ w be a subpath: we thus claim that | R | ≤ d Γ ( R − , R + ). Since P ′ v e Q and Qf P ′ w are geodesicsin Γ, we may assume, without loss of generality, that R + ∈ f P ′ w and R − ∈ P ′ v e . Moreover,assume that d Γ ( R + , Q + ) ≤ d Γ ( R − , Q − ): the other case is analogous. We then have d Γ ( R − , R + ) ≥ d Γ ( R − , Q + ) − d Γ ( R + , Q + )= d Γ ( R − , Q − ) + | Q | − d Γ ( R + , Q + ) ≥ | Q | since P ′ v e Q is a geodesic in Γ. On the other hand, since P ′ v e P v is a geodesic in Γ of length ≤ k ,we have | P ′ v e | = | P ′ v | + | e | ≤ k , and similarly | f P ′ w | ≤ k . It follows that | R | ≤ | P ′ v e Qf P ′ w | = | P ′ v e | + | Q | + | f P ′ w | ≤ | Q | + 2 k , and so, since | Q | ≥ k , we have | R | ≤ | Q | + 2 k ≤ | Q | ≤ d Γ ( R − , R + ), as claimed.But now, by the choice of k , it follows that P ′ v e Qf P ′ w is in the k -neighbourhood of W in Γ,and so, in particular, all the vertices of Q belong to ∆. Since P ′ v e P v and P ′ w f P w are geodesicsin Γ of length ≤ k , it also follows that all vertices of P v and of P w belong to ∆. Therefore, allthe vertices of P v e Qf P w belong to ∆. But as P v e Qf P w is a geodesic in Γ, there exists ageodesic in Γ k from v to w all of whose vertices are also vertices of P v e Qf P w , and so of ∆.This contradicts the choice of v and w .Therefore, ∆ must be an isometrically embedded subgraph of Γ k , as required. (cid:3) Proof of Theorem 1.4.
We first make the following elementary observation.
Claim.
Let Θ be a locally finite graph, and H a group acting on Θ . Suppose that there exists x ∈ V (Θ) such that | Stab H ( x ) | < ∞ and the H -orbit x · H is finite Hausdorff distance awayfrom V (Θ) . Then the H -action on Θ is geometric.Proof of Claim. Since Θ is locally finite, it is enough to show that the H -action is cocompactand | Stab H ( y ) | < ∞ for all y ∈ V (Θ).Let k < ∞ be the Hausdorff distance from x · H to V (Θ). If e ⊆ Θ is an edge, then thereexists h ∈ H such that d Θ ( e ′− , x ) ≤ k , and so d Θ ( e ′± , x ) ≤ k + 1, where e ′ = e · h . But sinceΘ is locally finite, there exist only finitely many edges e ′ ⊆ Θ with d Θ ( e ′± , x ) ≤ k + 1, and sothere are only finitely many of orbits of edges under the H -action on Θ. Thus the action iscocompact, as required.Now let y ∈ V (Θ). Then there exists h ∈ H such that d Θ ( y ′ , x ) ≤ k where y ′ = y · h ; inparticular, if g ∈ Stab H ( y ′ ) then d Θ ( y ′ , x · g ) = d Θ ( y ′ , x ) ≤ k , and so x · g ∈ B k ( y ′ ; Θ). Since Θis locally finite, this implies that | Stab H ( y ) | = | h Stab H ( y ′ ) h − | = | Stab H ( y ′ ) | ≤ | B k ( y ′ ; Θ) | × | Stab H ( x ) | < ∞ , as required. (cid:4) We now prove parts (i) and (ii) of the Theorem.(i) By assumption, there exists x ∈ V (Γ) such that the orbit x · H ⊆ V (Γ) is (5 , k ≥ , x · H is in the k -neighbourhood of x · H , and let ∆ be the full subgraph of Γ k spannedby S h ∈ H B ( x · h ; Γ k ). ELLY GROUPS, COARSE HELLY GROUPS, AND RELATIVE HYPERBOLICITY 29
Let Θ = Helly(∆) be the Hellyfication of ∆, as defined in [CCG + § +
20, Theorem 4.4], Θ is a Helly graph contained as a subgraph in every Hellygraph containing ∆ as an isometrically embedded subgraph. As Γ is Helly, it is alsopseudo-modular, and so by Lemma 5.4, ∆ is isometrically embedded in Γ k ; moreover,by Lemma 5.2, Γ k is Helly. It follows that Θ is (isomorphic to) a subgraph of Γ k . Butsince Γ is locally finite, so is Γ k and therefore so is Θ. Thus, Θ is a locally finite Hellygraph, and so it is enough to show that H acts on Θ geometrically.Now the G -action on Γ induces a G -action on Γ k , with respect to which ∆ is clearly H -invariant. Furthermore, the H -action on ∆ extends to an H -action on Θ: see [CCG + ξ = 0), ∆ is coarse Helly. It follows by[CCG +
20, Proposition 3.12] that V (Θ) is Hausdorff distance ℓ < ∞ away from V (∆),and so Hausdorff distance ≤ ℓ + 1 away from x · H (in Θ). Moreover, since the action of G on Γ, and so on Γ k , is geometric, we have | Stab H ( x ) | ≤ | Stab G ( x ) | < ∞ . It followsfrom the Claim that the action of H on Θ is geometric, as required.(ii) Let ξ ≥ ξ -coarse Helly. By assumption, there exists x ∈ V (Γ) suchthat the orbit W := x · H ⊆ V (Γ) is (1 , ξ )-quasiconvex. Let k ≥ ⊆ Γ k the subgraph given by Lemma 5.3. Since Γ is locally finite, so is Γ k ; moreover,the G -action on Γ induces a G -action on Γ k . By construction, the subgraph ∆ ⊆ Γ k is H -invariant and is Hausdorff distance ≤ H -orbit W = x · H . As asubgraph of Γ k , ∆ is also locally finite; furthermore, | Stab H ( x ) | ≤ | Stab G ( x ) | < ∞ . Itfollows from the Claim that the H -action on ∆ is geometric. But by Lemma 5.3, ∆ iscoarse Helly, as required. (cid:3) References [AC16] Y. Antol´ın and L. Ciobanu,
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Instytut Matematyczny, Uniwersytet Wroc lawski, plac Grunwaldzki 2/4, 50-384 Wroc law,Poland
Institute of Mathematics, Polish Academy of Sciences, ´Sniadeckich 8, 00-656 Warszawa, Poland
Email address : [email protected] Instytut Matematyczny, Uniwersytet Wroc lawski, plac Grunwaldzki 2/4, 50-384 Wroc law,Poland
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