Conformal dimension of hyperbolic groups that split over elementary subgroups
CCONFORMAL DIMENSION OF HYPERBOLIC GROUPS THATSPLIT OVER ELEMENTARY SUBGROUPS
MATIAS CARRASCO AND JOHN M. MACKAY
Abstract.
We study the (Ahlfors regular) conformal dimension of the bound-ary at infinity of Gromov hyperbolic groups which split over elementary sub-groups. If such a group is not virtually free, we show that the conformaldimension is equal to the maximal value of the conformal dimension of the ver-tex groups, or 1, whichever is greater, and we characterise when the conformaldimension is attained. As a consequence, we are able to characterise whichGromov hyperbolic groups (without 2-torsion) have conformal dimension 1,answering a question of Bonk and Kleiner. Introduction
Overview.
The conformal dimension of a metric space, introduced by Pansu,is the infimal Hausdorff dimension of all the quasisymmetrically equivalent met-rics on the space. It is a natural quasisymmetric invariant, and is connected tothe uniformisation problem of finding an optimal (“flattest”) metric for a givenspace. Since the boundary at infinity ∂ ∞ G of a Gromov hyperbolic group G car-ries a canonically defined family of metrics that are pairwise quasisymmetric, theconformal dimension of ∂ ∞ G is a well-defined quasi-isometric invariant of G . Theinitial motivation for the introduction of this invariant by Pansu in [Pan89] was inthe study of the large scale geometry of negatively curved homogeneous spaces, forwhich the conformal dimension can be computed explicitly. However, in generalit is an invariant that is very hard to compute. Despite this difficulty, it has foundapplications in other areas of geometric group theory and dynamical systems.These include the work of Bonk and Kleiner on the rigidity of quasi-M¨obius groupactions [BK02b]; the works of Bonk and Kleiner [BK05a] and Ha¨ıssinsky [Ha¨ı15]on Cannon’s conjecture and the boundary characterisation of Kleinian groups; theworks of Ha¨ıssinsky and Pilgrim on the characterisation of rational maps amongcoarse expanding conformal dynamical systems on the 2-sphere [HP14]; the worksof Bourdon and Kleiner focussing on the relations between the (cid:96) p -cohomology, theconformal dimension, combinatorial modulus, and the Combinatorial LoewnerProperty [BK13, BK15]; and the works of the second author on conformal dimen-sion bounds for small cancellation and random groups [Mac12, Mac16], as well asfurther connections to actions on L p -spaces [Bou16, DM19]. We refer the readerto the survey [MT10] for the basic theory of conformal dimension and its firstapplications. Date : July 20, 2020.1991
Mathematics Subject Classification.
Key words and phrases. conformal dimension, hyperbolic groups, graph of groupsdecomposition.This research was supported in part by EPSRC grant EP/P010245/1 and the MathAMSUDproject Geometry and Dynamics of Infinite Groups. a r X i v : . [ m a t h . M G ] J u l MATIAS CARRASCO AND JOHN M. MACKAY
In this paper we compute the conformal dimension of a hyperbolic group thatsplits as a graph of groups with elementary edge groups in terms of the conformaldimensions of the resulting vertex groups. Throughout the paper, an elementary(sub)group is a group that is finite or 2-ended, i.e., virtually Z . Unless otherwiseindicated, by ‘conformal dimension’ we mean the now more commonly studiedAhlfors regular conformal dimension, see Definition 3.1. Theorem 1.1.
Suppose G is a hyperbolic group, and we are given a graphs ofgroups decomposition of G , with vertex groups { G i } and all edge groups are ele-mentary. Then if G is not virtually free, Confdim ∂ ∞ G = max (cid:110) { } ∪ { Confdim ∂ ∞ G i : G i infinite } (cid:111) . This enables us to characterise those hyperbolic groups (under the mild assump-tion of no 2-torsion) which have conformal dimension equal to one, answering aquestion of Bonk and Kleiner [BK05a, Question 6.1], and giving new types ofexamples of self-similar metric spaces of conformal dimension 1.
Corollary 1.2. If G is a hyperbolic group with no -torsion and not virtuallyfree, then Confdim ∂ ∞ G = 1 if and only if G has a hierarchical decompositionover elementary edge groups so that each vertex group is elementary or virtuallyFuchsian. Let us now discuss these results in more detail. The case of Theorem 1.1 whenall the edge groups are finite is well-known in the field (a proof may be found inthe first author’s thesis [Car11, Theorem 6.2]).
Theorem 1.3. If G is an infinite hyperbolic group with a finite graph of groupsdecomposition where the vertex groups are { G i } and the edge groups are finite,then Confdim ∂ ∞ G = max { Confdim ∂ ∞ G i : G i infinite } , where max ∅ = 0 . In light of this result, Theorem 1.1 reduces to the following:
Theorem 1.4.
Suppose G is a hyperbolic group with a graph of groups decompo-sition of G with vertex groups { G i } and all edge groups -ended, then if G is notvirtually free, Confdim ∂ ∞ G = max (cid:110) { } ∪ { Confdim ∂ ∞ G i } (cid:111) . Proof of Theorem 1.1.
The lower bound for Confdim ∂ ∞ G is immediate: if wehave Confdim ∂ ∞ G < ∂ ∞ G = 0 and G is virtually free (see e.g.[MT10, Theorem 3.4.6]), thus Confdim G ≥
1. In addition, each G i is a quasicon-vex subgroup of G so each infinite G i has ∂ ∞ G i is quasisymmetrically embeddedin ∂ ∞ G , therefore Confdim ∂ ∞ G ≥ Confdim ∂ ∞ G i .For the upper bound, amalgamate all edges with infinite edge groups to get aless refined graph of groups decomposition G (cid:48) , where the conformal dimension ofthe new vertex groups has the bound from Theorem 1.4. Then as all edge groupsin G (cid:48) are finite, the upper bound follows from Theorem 1.3. (cid:3) Particular cases of Theorems 1.1 and 1.4 were known before. Keith and Kleinerin unpublished work [KK] and Carrasco [Car14] showed that if ∂ ∞ G has wellspread local cut points (“WS” for short), then ∂ ∞ G has conformal dimension 1. ONFORMAL DIMENSION AND SPLITTINGS 3
By saying ∂ ∞ G has WS we mean that for some (any) fixed metric in the family,for any δ > ∂ ∞ G so that all remainingconnected components have diameter at most δ .As Theorem 1.1 applies whether WS holds or not, we can complete the “if”direction of Corollary 1.2 characterising which hyperbolic groups have conformaldimension one. The “only if” direction of Corollary 1.2 follows from work of thesecond author showing that hyperbolic groups with, for example, Sierpi´nski carpetor Menger sponge boundaries have conformal dimension greater than one, and anaccessibility result of Louder–Touikan [LT17]. Proof of Corollary 1.2.
Suppose G admits a finite hierarchy of graph of groupsdecompositions over finite and 2-ended subgroups, ending with vertex groups thatare elementary or virtually Fuchsian; such groups have conformal dimension atmost 1. Since G is not virtually free we have Confdim G ≥
1, and by repeatedlyapplying Theorem 1.1 we have that Confdim G ≤ ∂ ∞ G = 1. As G has no 2-torsion,[LT17, Corollary 2.7] implies that we can find a finite hierarchy for G as follows:by Stallings and Dunwoody we can split G maximally over finite edge groupsleaving finite or one-ended vertex groups, then take the JSJ decomposition of theone-ended (hyperbolic) vertex groups, maximally splitting over 2-ended subgroups,then repeat the Stallings–Dunwoody spliting for any vertex group with more thanone end, and so on, repeating finitely many times until all the vertex groupsremaining are either elementary, virtually Fuchsian groups, or one-ended groupsthat do not split over a 2-ended subgroup.Each vertex group is quasiconvex in the original group G as all splittings wereover elementary subgroups. The third case of one-ended, non-virtually Fuchsiangroups with no splittings over a virtually Z subgroup cannot arise, as such groupshave conformal dimension > (cid:3) Remark 1.5.
Corollary 1.2 holds also with the definition of conformal dimensionas the infimal Hausdorff dimension of (not necessarily Ahlfors regular) quasisym-metrically equivalent metrics; let us denote this by
Confdim H . First, if G admitssuch a hierarchy and is not virtually free, ≤ Confdim H ∂ ∞ G ≤ Confdim ∂ ∞ G =1 . Second, as the lower bound > from [Mac10] works for Confdim H also, if Confdim H ∂ ∞ G = 1 then all vertex groups in the hierarchical decomposition mustbe elementary or virtually Fuchsian as desired. Remark 1.6.
The groups considered in Corollary 1.2, when torsion free, are thegroups Wise suggests might be the hyperbolic virtual limit groups [Wis18, Section1.4] . Attainment of conformal dimension.
It is natural to ask when the con-formal dimension of a hyperbolic group is attained, i.e. when ∂ ∞ G is quasisym-metric to an Ahlfors Q -regular space with Q = Confdim ∂ ∞ G . When this issatisfied G often has rigidity properties, see the results and discussion in [Kle06].Under the hypothesis of Corollary 1.2, Bonk and Kleiner have shown that ifa hyperbolic group G has Confdim ∂ ∞ G = 1 and this is attained, i.e. if ∂ ∞ G is quasisymmetric to an Ahlfors 1-regular space, then ∂ ∞ G is a circle and G isvirtually Fuchsian [BK02b, Theorem 1.1].When we have a graph of groups as in Theorem 1.1, we can show the following. MATIAS CARRASCO AND JOHN M. MACKAY
Theorem 1.7.
Suppose G is a hyperbolic group, and we are given a graph ofgroups decomposition of G with vertex groups { G i } and all edge groups elementary.Then the conformal dimension of ∂ ∞ G is attained if and only if either: • Confdim ∂ ∞ G = 0 and G is -ended, or • Confdim ∂ ∞ G = 1 and G is virtually cocompact Fuchsian, or • G = G i for some vertex group with ∂ ∞ G i attaining its conformal dimension Confdim ∂ ∞ G i > . The main idea here is that if the conformal dimension Confdim ∂ ∞ G is attained,then any “porous” subset has strictly smaller conformal dimension. Since, byTheorem 1.1, Confdim ∂ ∞ G = Confdim ∂ ∞ G i for some vertex group G i , and G i is a quasiconvex subgroup of G , the limit set Λ G i cannot be porous in ∂ ∞ G andone can conclude that G i must equal G .1.3. Idea of proof and toy example.
By work of Keith–Kleiner, Bourdon–Kleiner and the first author [KK, BK13, Car13], the (Ahlfors regular) conformaldimension of the boundary of a hyperbolic group X = ∂ ∞ G is equal to the crit-ical exponent of the combinatorial modulus of the family of all curves in X ofdiameter at least δ , for some fixed small δ . Prior to the works just cited, other au-thors who have used combinatorial modulus to study conformal dimension includePansu [Pan89] and Keith–Laakso [KL04]; see [Car13] for further discussion.These notions are formally defined in Section 3, but we can illustrate the ideahere with a toy example. Consider the double G = π ( S ) ∗ Z π ( S ) where S is aclosed surface of genus 2, and Z corresponds to a closed geodesic curve γ in S .The boundary ∂ ∞ G is (speaking informally) the limit of a countable collection ofcircles, corresponding to ∂ ∞ π ( S ), glued at pairs of points, corresponding to ∂ ∞ Z ,in a tree-like fashion given by the Bass–Serre tree of the splitting.The topological properties of the boundary depend on the type of curve γ chosen. If γ is a simple closed curve, then Pansu observed that Confdim ∂ ∞ G = 1by varying the hyperbolic structure on S to find CAT( −
1) model spaces for G withvolume entropy arbitrarily close to that of the hyperbolic plane; see discussionin [BK05a, Section 6] and [Buy05, Theorem 1.1].If γ is not simple, but not filling, one cannot use this argument. Recall thata curve γ is filling if all connected components in S \ γ are topological discs, seeFigure 1 for an example of a filling curve. However, the second author observedthat such boundaries still satisfy the WS property, with cut points arising fromlimit points corresponding to an essential curve in S \ γ , and so Confdim ∂ ∞ G = 1here also. For a complete characterisation of when ∂ ∞ G has WS, including thiscase, see the work of the first author [Car14, Theorem 1.3].The case when γ is filling remained unresolved, but now we can apply Theo-rem 1.1 to find that Confdim ∂ ∞ G = 1.To show how to prove this, we sketch the idea for a toy example which models ∂ ∞ G . We build the space in stages, beginning by letting X be a circle with lengthmetric and of diameter 1, and fix two antipodal points x − , x + ∈ X . Define X bytaking X and gluing on at pairs of points on say 12 copies of X scaled down by1 / X in an overlapping fashion. For each n = 2 , , . . . , define X n in the following way: take a copy of S , and for each j = 1 , . . . , n glue on at pairsof points between 3 j − and 12 · j − copies of X n − j scaled down by 1 / j , spacedaround S . We assume that these gluings are done in a self-similar way, so there is ONFORMAL DIMENSION AND SPLITTINGS 5
Figure 1.
Some lifts in the hyperbolic plane of the filling curve abcd of the surface group (cid:104) a, b, c, d | [ a, b ][ c, d ] = 1 (cid:105) . x − x + Figure 2.
A toy model for the boundary of a surface groupdoubled along a filling curvea natural limit space X of this construction; see Figure 2 for a partial illustrationof how X is constructed. In the figure, circles are coloured black, blue, red, green.While the circles appear to overlap, a circle coloured blue, red or green meets noother circle of the same colour, and exactly one circle of some preceeding colourat exactly one pair of points.To show that Confdim X = 1, since Confdim X ≥ dim top X = 1 is trivial, itsuffices to show that Confdim X ≤ p for an arbitrary p >
1. Using the machineryof Keith–Kleiner and Carrasco mentioned above, such a bound follows from acombinatorial modulus estimate on X . Rather than considering all curves in X MATIAS CARRASCO AND JOHN M. MACKAY
Figure 3.
A cartoon of the weight function ρ of diameter ≥ δ , we simplify the argument here by considering the family of allpaths in X joining x − to x + , which we call Γ.For each n ∈ N , let S n be the cover of X by sets of size 3 − n corresponding to thecopies of X of size 3 − n in X n . A weight function ρ n : S n → (0 , ∞ ) is admissible for Γ if for any γ ∈ Γ, the ρ n -length (cid:96) ρ n ( γ ) satisfies (cid:96) ρ n ( γ ) := (cid:88) A ∈S n : γ ∩ A (cid:54) = ∅ ρ n ( A ) ≥ . Roughly speaking, a weight function describes a hoped for conformal deformationwhere the desired diameters of the images of A ∈ S n are the values of ρ n ( A ), andadmissibility ensures that the image doesn’t collapse down in size. The p -volume Vol p ( ρ n ) of ρ n is defined as a n := Vol p ( ρ n ) := (cid:88) A ∈S n ρ n ( A ) p . To achieve the bound Confdim X ≤ p , we require a sequence ( ρ n ) of Γ-admissibleweight functions so that a n = Vol p ( ρ n ) → n → ∞ .We now define ρ n : S n → (0 , ∞ ) and estimate a n by induction. The first step iseasy: S = { A } is a cover of X by a single open set, and we let ρ : S → (0 , ∞ ), ρ ( A ) = 1, which is admissible and has a := 1.Now for the inductive step: assume that suitable ρ i have been defined for all i = 0 , . . . , n −
1. The idea at step n is that we send the geometric sequence ofannuli A − i := B ( x − , − i / \ B ( x − , − ( i +1) / i = 0 , . . . , n −
1, to an arithmeticsequence of annuli each of size 1 / n , and likewise for the annuli A + i centred at x + .This will define an admissible weight function; see Figure 3 for an illustration.Now, we describe ρ n in more detail (though not with an explicit formula), andwe estimate its p -volume a n . For each i = 0 , . . . , n −
1, and each j = i + 1 , . . . , n ,the annuli A − i , A + i contain in total ≤ C − i / − j = C j − i copies of X n − j , which weendow with weights using ρ n − j ; here C > ≥ /n in the image, we apply a scaling factor of 1 / (3 j − i n )to these copies, which scales a n − j by 1 / (3 j − i n ) p . Thus, summing these up andusing geometric series bounds, we have a n ≤ C n − (cid:88) i =0 n (cid:88) j = i +1 j − i · a n − j (3 j − i n ) p = Cn p n (cid:88) j =1 j − (cid:88) i =0 − ( j − i )( p − a n − j ≤ C (cid:48) n p n (cid:88) j =1 a n − j ≤ C (cid:48) n p − max { a , . . . , a n − } , ONFORMAL DIMENSION AND SPLITTINGS 7 for some constant C (cid:48) >
1. This inequality implies that for n large, the sequence( a n ) is nonincreasing, hence ( a n ) is bounded, hence the inequality again impliesthat a n → n → ∞ . The proof is complete.The general argument in the paper is more involved in several ways, but hasthe same key idea of deforming geometric sequences of annuli into arithmeticsequences at its foundation. Many additional complications are laid on by incor-porating deformations of ∂ ∞ G i which nearly achieve the conformal dimension ofthe boundaries of each space ∂ ∞ G i , carefully checking admissibility (for all curvesof given diameter, not just those joining two points), and setting up a suitableinduction for the volume bounds.1.4. Outline of paper.
In Section 2 we describe the metric properties of limitsets in hyperbolic groups with quasiconvex splittings. In Sections 3-7 we proveTheorem 1.4: Section 3 reduces the theorem to a statement about combinatorialmodulus, and in Section 4 a sequence of weight functions is defined. The weightfunctions are shown to have maximum values going to zero, to be admissible, andto have bounded volume in Sections 5, 6, and 7 respectively. Finally, in Section 8we consider attainment of conformal dimension and prove Theorem 1.7.1.5.
Notation.
We write A (cid:22) B if A ≤ CB for some constant C >
0, and A (cid:16) B if A (cid:22) B and B (cid:22) A . We may write A (cid:22) C B or A (cid:16) C B to indicate which C .We also write A ≈ B if | A − B | ≤ C for some constant C ≥
0. Throughout thepaper,
C, C (cid:48) , C (cid:48)(cid:48) , . . . , C , C , . . . refer to constants only depending on the relevantdata; sometimes we make the dependence clear by writing C = C ( α, β, ... ) and soon. For A, B ∈ R , we write A ∨ B := max { A, B } and A ∧ B := min { A, B } .1.6. Acknowledgements.
The first author thanks Peter Ha¨ıssinsky who firstintroduced him to this question many years ago. The second author thanks BruceKleiner and Daniel Meyer for helpful conversations about this question over theyears. Thanks go to Daniel Groves, Lars Louder and Henry Wilton for helpfulcomments.2.
Graph of groups decompositions and boundaries
In this section we present useful facts about boundaries and quasiconvex split-tings of hyperbolic groups that will be used later. For references on graph ofgroups and Bass–Serre theory, see Serre [Ser03], Scott–Wall [SW79] and Drut¸u–Kapovich [DK18].2.1.
Boundaries of quasiconvex splittings. An abstract (oriented) graph G consists of two sets, the vertices V G and the edges E G , with an initial vertex map( · ) − : E G → V G , e (cid:55)→ e − and a terminal vertex map ( · ) + : E G → V G , e (cid:55)→ e + .Suppose G acts on a tree T without inversions on edges, minimally (i.e. thereis no proper invariant sub-tree of T ), and with the quotient graph G \ T finite.Any such action corresponds to a graph of groups decomposition G for G where theunderlying graph is G \ T , for each vertex i ∈ V ( G \ T ) the vertex group is a copyof the stabilizer G v for some v ∈ V T in the orbit corresponding to i , for each edge k ∈ E ( G \ T ) the edge group is a copy of the stabilizer G e for some e ∈ ET in theorbit corresponding to k , and the injective homomorphisms from edge groups intovertex groups are induced by the inclusions of stabilizers G e → G e − , G e → G e + .We call T the Bass–Serre tree for the graph of groups decomposition G . MATIAS CARRASCO AND JOHN M. MACKAY
As all the stabilizers in an orbit are conjugate, for v ∈ V T we can define i v ∈ V G , g v ∈ G so that G v = g v G i v g − v , and for e ∈ ET we can define k e ∈ E G , g e ∈ G so that G e = g e G k e g − e .We now build a model space Z for G . For each i ∈ V G let M i be a presentationcomplex for G i , so M i is a 2-dimensional cell complex with π ( M i ) = G i . Likewisefor each k ∈ E G let M k be a presentation complex for G k . The homomorphismsfrom edge groups to vertex groups are induced by continuous maps f k − : M k → M k − , f k + : M k → M k + for k ∈ E G . The graph of spaces M is built from thecollection { M i } i ∈ V G ∪ { M k × [ − , } k ∈ E G where we glue each M k × {± } to M k ± by the map ( z, ± (cid:55)→ f k ± ( z ). By Bass–Serre theory, the fundamental group π ( M ) equals G .Define a length metric on M which induces a geodesic metric on the universalcover Z := (cid:102) M . This space Z is a tree of spaces with a copy Z v of (cid:103) M i v for each v ∈ V T and a copy Z e × [ − ,
1] of (cid:103) M k e × [ − ,
1] for each e ∈ ET , where the subset (cid:103) M k e ×{± } is glued into the corresponding vertex spaces. The action G (cid:121) Z = (cid:102) M preserves this tree-of-spaces structure, and so if we collapse each vertex space Z v to a point and each edge space Z e × [ − ,
1] to an edge we recover our original tree T and action G (cid:121) T .Fix a base vertex v ∈ T and a basepoint o ∈ Z so that Z → T maps o to v .As G acts geometrically on Z the orbit map G → G · o induces a quasi-isometry G → Z . This quasi-isometry coarsely maps each left coset g v G i v , v ∈ V T to Z v ,and likewise coarsely maps each g e G k e , e ∈ ET , to Z e × [ − , G is a hyperbolic group, and so Z is hyperbolic also. We fix a visualmetric d on X := ∂ ∞ Z with visual parameter (cid:15) >
0, i.e. d ( · , · ) (cid:16) e − (cid:15) ( ·|· ) o , where( ·|· ) o denotes the Gromov product with basepoint o . We may rescale to assumediam X = 1.For a subgroup H of G , let Λ( H ) ⊂ X be the limit set of H , i.e. the accumulationpoints of H · o in X = ∂ ∞ Z . For v ∈ V T we denote the limit set of the stabilizer G v by Λ v = Λ( G v ), and likewise for e ∈ ET we let Λ e = Λ( G e ).In each case considered here, the edge groups are uniformly quasiconvex as theyare either finite or two-ended. Therefore the vertex groups are uniformly quasicon-vex also (see e.g. [Bow98, Proposition 1.2]), and so hyperbolic, and consequentlyfor each v ∈ V T the quasi-isometric embedding g v G i v → Z found by restrictingthe orbit map induces a quasisymmetry ∂ ∞ g v G i v = g v ∂ ∞ G i v → Λ v ⊂ ∂ ∞ Z . Lemma 2.1 (cf. [Bow98, Proposition 1.3], [KK00, Lemma 10]) . If G is a hyper-bolic group with a graph of groups decomposition G over quasiconvex edge groupswith Bass–Serre tree T , with G acting geometrically on the model space Z , and X = ∂ ∞ Z with a visual metric, then every x ∈ X corresponds to exactly one ofthe following: • a point of ∂ ∞ T , with a unique x for each t ∈ ∂ ∞ T , or • a point of Λ e for some e ∈ ET , or • a point of Λ v for some unique v ∈ V T (but not in any Λ e ).Proof. Consider a geodesic ray γ from o in Z representing x ∈ X .For an edge e ∈ ET let Z e → be the component of Z \ ( Z e × { } ) not containing o . Let us say that γ essentially crosses the edge space corresponding to e ∈ ET ,or just γ essentially crosses e , if for every C > γ has unbounded intersectionwith Z e → \ N C ( Z e × { } ). ONFORMAL DIMENSION AND SPLITTINGS 9 If γ essentially crosses e ∈ ET , then it essentially crosses every edge between v and e in T . Moreover if a simple path from v to some vertex v in T can beextended by either e (cid:48) or e (cid:48)(cid:48) in ET , then by quasiconvexity γ cannot essentiallycross both e (cid:48) and e (cid:48)(cid:48) . Therefore the collection of edges in T which γ essentiallycrosses gives a simple path from v , either (i) infinite or (ii) finite. Let us call thispath ˆ γ : by definition it depends only on the point x ∈ X and not the choice of γ .In case (i), the path ˆ γ determines a unique point in ∂ ∞ T . We claim that thereis a bijection between the set of x ∈ X represented by γ with ˆ γ unbounded, andpoints in ∂ ∞ T . First, given any point t ∈ ∂ ∞ T , by an Arzel`a–Ascoli argumentone can choose a geodesic ray γ in Z so that ˆ γ limits to t .Second, if γ, α are geodesic rays and ˆ γ = ˆ α is unbounded, then γ and α mustrepresent the same point in X : suppose not, then ( γ | α ) o < ∞ . Choose a largeconstant R and an edge e ∈ ET which γ and α essentially cross so that the edgespace Z e × { } is outside B ( o, ( γ | α ) o + R ). Let p, q ∈ Z e × { } be points where γ and α respectively meet the edge space. By hyperbolicity, the geodesic from p to q must go within distance ( γ | α ) o + C of o , but by quasiconvexity it mustremain within a distance C of Z e × { } , a contradiction for R > C . So case (i)is understood.Now suppose we are in case (ii), where ˆ γ is a finite path with final vertex v ,and final edge e . If γ leaves Z v through some Z e (cid:48) × { } and does not return, as itdoes not essentially cross e (cid:48) it must limit to a point of Λ e (cid:48) . So if γ does not limitto a point of any edge space, by quasiconvexity its tail must live in N C ( Z v ) forsome constant C , and so x ∈ Λ v . If x ∈ Λ v (cid:48) also for some v (cid:48) (cid:54) = v , then the tail of γ must live in N C ( Z v (cid:48) ) also, hence in N C ( Z e (cid:48) ) for any edge e (cid:48) between v and v (cid:48) ;as this contradicts our assumption on γ we have that v is unique as required. (cid:3) In the rest of this section we will use the approximate self-similarity of theboundary of a hyperbolic group: there exists L ≥ x ∈ X = ∂ ∞ Z ,and all 0 < r ≤ diam X , there exists g ∈ G so that the action of g on X inducesan L -bi-Lipschitz map from the rescaled ball ( B ( x, r ) , r d ) to an open set U ⊂ X with B ( gx, L ) ⊂ U . Lemma 2.2 (cf. [BL07, Proposition 6.2], [BK13, Proposition 3.3], [MS19, Corol-lary 4.9]) . Suppose Z is a hyperbolic, proper, geodesic metric space with a geo-metric group action G (cid:121) Z , base point o , and a visual metric d on X = ∂ ∞ Z with visual parameter (cid:15) . Then there exists L ≥ so that X is approximatelyself-similar.Proof. By the cocompactness of G (cid:121) Z there exists D > G · B Z ( o, D ) = Z . Let L be given by [MS19, Corollary 4.9] applied to D , the hyperbolicityconstant δ Z for Z , and parameters C , (cid:15) for the visual metric d .Suppose we are given x ∈ X and r ∈ (0 , diam X ]. If − (cid:15) − log(2 rC ) − δ Z − ≥ y ”=“ x ”, “ r (cid:48) ”= r , and an appropriate g ∈ G gives an L (cid:48) -bi-Lipschitz map from ( B ( x, r ) , r ) to an open set U ⊂ X with B ( f ( x ) , L (cid:48) ) ⊂ U . Otherwise, − (cid:15) − log(2 rC ) − δ Z − < ∈ G gives approximate self-similarity. (cid:3) Connected components in boundaries.
Maximal splittings over finiteedge groups enable us to control the geometry of connected components in any space arising as the boundary of any space admitting a geometric action by ahyperbolic group.Recall that a metric space X is C -linearly connected for some C ≥ x, y ∈ X there is a compact connected set I ⊂ X with diam I ≤ Cd ( x, y ). The following definition is used in the proof of Theorem 3.2. Definition 2.3 (see [Car13, Theorem 3.11]) . The components of a metric spaceare uniformly linearly connected if they are each K (cid:96) -linearly connected for somefixed K (cid:96) ≥ .The components are uniformly separated if for some fixed K s ≥ , for all < r ≤ diam X , there exists a covering W r of X , by open and closed sets, suchthat for all W ∈ W r , we have d ( W, X \ W ) ≥ r/K s and there exists a connectedcomponent Y of X with Y ⊂ W and W is contained in the r -neighbourhood of Y . Recall that by Stallings–Dunwoody [Sta68, Dun85], there is a maximal graph ofgroups decomposition of G where all edge groups are finite, and the vertex groups { G i } are all finite or one-ended. Lemma 2.4.
Suppose G is a hyperbolic group acting geometrically on a geodesicspace Z (cid:48) , and let X (cid:48) = ∂ ∞ Z (cid:48) with a fixed visual metric d (cid:48) . Let T be a Bass-Serre tree corresponding to a Stallings–Dunwoody decomposition for G with vertexstabilizers denoted { G v } . Then (1) the connected components of X (cid:48) correspond to Λ v for G v one-ended, andto points in ∂ ∞ T ; (2) X (cid:48) has uniform linear connectivity of components; (3) X (cid:48) has uniform separation of components. The uniform separation of components condition is tricky to work if we let X (cid:48) be arbitrarily quasisymmetrically equivalent to X , but we only need the case ofvisual metrics d (cid:48) as in the lemma. Proof.
Consider a Stallings–Dunwoody graph of groups decomposition of G withcorresponding tree T . Let Z be the model space for this graph of groups decompo-sition for G constructed as in the previous section with base point o , let X = ∂ ∞ Z be its boundary with a visual metric d as before.Since G acts geometrically on Z (cid:48) , there is a quasi-isometry φ : Z → Z (cid:48) whichsends the orbit G · o to the orbit G · φ ( o ) equivariantly; let o (cid:48) := φ ( o ). Let ψ : Z (cid:48) → Z be a quasi-inverse of φ , sending G · o (cid:48) to G · o equivariantly. Asbefore, write Λ v = Λ( G v ) , Λ e = Λ( G e ) for the given limit sets in X , and letΛ (cid:48) v , Λ (cid:48) e be the corresponding limit sets in X (cid:48) . By equivariance, ∂ ∞ φ (Λ v ) = Λ (cid:48) v and ∂ ∞ φ (Λ e ) = Λ (cid:48) e .We now begin the proof of (1). Since the edge groups are finite, in Z the edgespaces Z e × { } , e ∈ ET , have uniformly bounded diameter.Consider a geodesic ray γ (cid:48) from o (cid:48) in Z (cid:48) . Let γ be a geodesic ray from o in Z atbounded Hausdorff distance from the quasi-geodesic ψ ( γ (cid:48) ). By Lemma 2.1, andsince Λ e = ∅ for any e ∈ ET , every x ∈ X either corresponds to a point of ∂ ∞ T ,or to a point in Λ v for a unique v ∈ V T .We define the simple path ˆ γ in T as in the proof of Lemma 2.1; by constructionit is independent of the choice of γ , so we write ˆ γ (cid:48) := ˆ γ . Suppose we have twogeodesic rays γ (cid:48) , α (cid:48) in Z (cid:48) with corresponding geodesic rays γ, α in Z as above. If γ essentially crosses some edge e ∈ ET and α does not, then the Gromov product ONFORMAL DIMENSION AND SPLITTINGS 11 ( γ | α ) o is ≤ d Z ( o, Z e × { } ) + C , and so the condition “essentially crossing e ”splits X into two sets at positive distance, thus the limit points of γ and α arein different connected components of X . As ∂ ∞ φ is a homeomorphism, the limitpoints of γ (cid:48) and α (cid:48) are in different connected components of X (cid:48) . Thus if the limitpoints of γ (cid:48) and α (cid:48) are in the same connected component of X (cid:48) , then ˆ γ (cid:48) = ˆ α (cid:48) .If ˆ γ (cid:48) is unbounded, and α (cid:48) is a geodesic ray in Z (cid:48) , either ˆ α (cid:48) (cid:54) = ˆ γ (cid:48) and so α (cid:48) limitsto a different connected component of X (cid:48) , or ˆ α (cid:48) = ˆ γ (cid:48) , so by Lemma 2.1 α and γ represent the same point in X = ∂ ∞ Z , and so α (cid:48) and γ (cid:48) represent the same pointin X (cid:48) = ∂ ∞ φ ( X ). This point corresponds to the point in ∂ ∞ T represented by ˆ γ (cid:48) .On the other hand, if ˆ γ (cid:48) is a finite path, let v be the final vertex of the path.Therefore γ must meet Z v in an unbounded set, and limits to a point of Λ v , andso γ (cid:48) limits to a point of Λ (cid:48) v , which is the image of ∂ ∞ g v G i v under the boundaryof the orbit map G → Z (cid:48) . As G i v is infinite it is one-ended, so ∂ ∞ G i v is connectedand thus so is Λ (cid:48) v , and every geodesic ray α (cid:48) with ˆ α (cid:48) = ˆ γ (cid:48) is in the same connectedset Λ (cid:48) v . So (1) is proved.We now prove (2). By Bonk–Kleiner [BK05b] the boundary of a one-endedhyperbolic group is linearly connected. If v ∈ V T corresponds to a one-endedvertex group, then as g v G i v → Z v → φ ( Z v ) is a quasi-isometry embedding into Z (cid:48) , the boundary map g v ∂ ∞ G i v = ∂ ∞ g v G i v → Λ v → Λ (cid:48) v is a quasisymmetricembedding, so Λ (cid:48) v is also linearly connected, though not a priori with constantsindependent of v .However, we can use the approximate self-similarity of X (cid:48) = ∂ ∞ Z (cid:48) . Let (cid:15) (cid:48) denotethe visual parameter of the metric d (cid:48) . For v ∈ V T we have diam Λ (cid:48) v (cid:22) e − (cid:15) (cid:48) d Z (cid:48) ( o (cid:48) ,φZ v ) because all geodesic rays from o (cid:48) to points in Λ (cid:48) v must pass within bounded distanceof the same edge space adjacent to φZ v , and in particular their Gromov productswith each other are all ≥ d Z (cid:48) ( o (cid:48) , φZ v ) − C . By Lemma 2.2, for all v we can find a g ∈ G so that, up to scaling, Λ (cid:48) v is uniformly bi-Lipschitz to g · Λ (cid:48) v = Λ (cid:48) gv , wherediam Λ (cid:48) gv ≥ /C >
0. That is, Λ (cid:48) gv is one of finitely many possible candidates.Thus the linear connectivity constant of Λ (cid:48) v may be taken independent of v . Wehave proven (2).It remains to show (3).Given R >
0, let E R be the set of edges of T so that the corresponding edgespaces φ ( Z e ) are within distance R of the base point o (cid:48) ∈ Z (cid:48) . Partition theboundary X (cid:48) according to the last edge in E R which the corresponding geodesicrays γ (cid:48) essentially cross, i.e., a geodesic representative γ of the quasi-geodesic ray ψ ( γ (cid:48) ) essentially crosses the edge. Denote the partition by W R .Notice that there is a set in this partition that corresponds to the rays whoseclass does not essentially cross any edge in E R . This set is a neighborhood of Λ (cid:48) v .The sets in this partition are closed since a limit of rays in the same set is also inthe same set. Since the partition is finite the sets are open as well.For W ∈ W R if x ∈ X (cid:48) \ W and w ∈ W then by the definition of W R we musthave ( x | w ) o (cid:48) ≤ R + C and so d (cid:48) ( x, w ) ≥ e − (cid:15) (cid:48) R /C (cid:48) for some constant C (cid:48) .Consider W ∈ W R corresponding to geodesic rays which essentially cross anedge e ∈ E R last, and let v be the vertex of e furthest from v . If W correspondsto rays not essentially crossing any edge of E R , let v = v .If G v is finite, then the (bounded) edge space for e is at distance ≥ R − C from o (cid:48) , else we would have to essentially cross another edge of E R . So we can take as Y ⊂ X (cid:48) the connected component of some such geodesic ray γ (cid:48) in W . Indeed, by the proof of (1) above, every geodesic ray corresponding to a point of Y essentiallycrosses the same edge e as γ (cid:48) , so Y ⊂ W . Also, if w ∈ W then for any y ∈ Y wehave ( y | w ) o (cid:48) ≥ R − C and so d (cid:48) ( y, w ) ≤ e − (cid:15) (cid:48) R C (cid:48) .On the other hand, if G v is infinite, let Y = Λ (cid:48) v ⊂ W . If w ∈ W then thelast point z of a geodesic ray to w in B ( o (cid:48) , R ) is (within bounded distance of)some point in φZ v , as the ray essentially crosses the same edges of E R as anygeodesic from o (cid:48) to φZ v . Since G v is infinite there is a geodesic line close to φZ v passing within bounded distance of z ; let y, y (cid:48) ∈ Y = Λ (cid:48) v be the limit points ofthe line. By hyperbolicity, one of the geodesic rays from o (cid:48) to y or to y (cid:48) passeswithin a uniformly bounded distance of z , so max { ( w | y ) o (cid:48) , ( w | y (cid:48) ) o (cid:48) } ≥ R − C , thus d (cid:48) ( Y, w ) ≤ e − (cid:15) (cid:48) R C (cid:48) .In conclusion the uniform separation of components, statement (3) of the lemma,is satisfied for K s := ( C (cid:48) ) and by taking W r := W R for R = − (cid:15) (cid:48) log( r/C (cid:48) ). (cid:3) Two-ended edge groups.
We now consider in more detail the case when alledge groups are two-ended (and hence all vertex groups are infinite). In general,stabilizers of different edge groups can have the same limit sets, so to get strongerresults about the geometry of such groups we switch from the given graph ofgroups to a new, bipartite, graph of groups. We follow Guirardel–Levitt [GL11].
Proposition 2.5.
Given a hyperbolic group G with a graph of groups decomposi-tion over -ended edge groups, we can find a graph of groups corresponding to anaction G (cid:121) T where the tree is bipartite with V T = V T (cid:116) V T , and (1) all V T vertex groups are non-elementary and are conjugate to some orig-inal vertex group; (2) all V T groups and all edge groups are -ended; (3) different V T vertex groups are not commensurable, and hence have disjointlimit sets in G ; (4) every original vertex group that was non-elementary is also a new V T vertex group.Proof. Let G (cid:121) S be the original tree action. The new tree T is what Guirardeland Levitt call the tree of cylinders of S . Their construction is as follows (fordetails see [GL11, Section 4], for an example see Figure 4).Define an equivalence relation ∼ S on the set of non-oriented edges of S by e ∼ S e (cid:48) if G e and G e (cid:48) are commensurable (i.e. if G e ∩ G e (cid:48) has finite index in both G e and G e (cid:48) ). A cylinder of S is an equivalence class [ e ] S .Notice that since G is hyperbolic, given two edges e and e (cid:48) of S either Λ e ∩ Λ e (cid:48) = ∅ or Λ e = Λ e (cid:48) . Moreover, e ∼ S e (cid:48) if and only if Λ e = Λ e (cid:48) .By [GL11, Lemma 4.2] every cylinder of S is connected, and hence a subtree.Since there are only finitely many conjugacy classes of edge groups, and there areonly finitely many conjugate edge groups that can contain a given loxodromic,every cylinder is finite.The tree of cylinders T is the bipartite tree with vertex set V T = V T (cid:116) V T defined as follows:(1) V T is the set of vertices v of S belonging to at least two distinct cylinders;(2) V T is the set of cylinders [ e ] S of S ;(3) and there is an edge between v and [ e ] S if v as a vertex of S belongs tothe union of edges of [ e ] S . ONFORMAL DIMENSION AND SPLITTINGS 13
CA Ba (cid:55)→ c (cid:55)→ b C C A B (cid:104) b (cid:105) a (cid:55)→ c (cid:55)→ b b (cid:55)→ c (cid:55)→ b (cid:104) b (cid:105)(cid:104) b (cid:105) A BbAb − (cid:104) b (cid:105) (cid:104) b (cid:105)(cid:104) b (cid:105) A bAb − (cid:104) b (cid:105) B Figure 4.
Let A = (cid:104) a , a (cid:105) , B = (cid:104) b , b (cid:105) be two copies of the freegroup and C = (cid:104) c (cid:105) (cid:39) Z . Let a = [ a , a ] and b = [ b , b ], andconsider the amalgamated product G = A ∗ C B where the injectionmaps are given by c (cid:55)→ a and c (cid:55)→ b . The group G is isomor-phic to the fundamental group of the complex obtained by gluingtwo punctured tori to a Mobius band, one of them glued along itsboundary to the boundary of the band, and the other one gluedalong its boundary to the mid-circle of the band. The splitting A ∗ C B corresponds to the graph of groups decomposition shownon the left. Notice that the edges of S issuing from a B -vertex arenaturally paired: in this case the cylinders are the pairs of edgeshaving the same stabilizer. The tree of cylinders T corresponds toreplacing these pairs by a tripod. The associated graph of groupsis shown on the right.That is, the tree T is obtained from S by replacing each cylinder by the coneon its boundary. See [Gui04, Definition 4.8] for the proof that T is indeed atree. Moreover, the group G acts on T and the action G (cid:121) T is also minimal,[Gui04, Lemma 4.9].Notice that a non-elementary stabilizer G v of a vertex v of S has infinite degreein S . Therefore, if a vertex of S belongs to only one cylinder, it has finite degreeand its stabilizer must be two-ended. That is, vertices of S with non-elementarystabilizers are also vertices in V T . This shows (4). Moreover, the stabilizer of avertex in V T is the same as the stabilizer of the corresponding vertex in S , so nonew non-elementary vertices are created by this construction, and this gives (1).The stabilizer of a vertex in V T is the global stabilizer of a cylinder [ e ] S in S , which coincides with the maximal two-ended subgroup containing G e (cid:48) for anyedge e (cid:48) ∈ [ e ] S . This proves the first claim of (2), and shows that an edge stabilizerof T is elementary. But if ( v, [ e ] S ) is an edge of T , then its stabilizer contains thestabilizer of the edge of [ e ] S incident to v , which is two-ended. Therefore edge stabilizers of S are two-ended (see also [GL11, Proposition 6.1]). This completesthe proof of (2). Property (3) follows directly by the definition of cylinders. (cid:3) Metric estimates for the limit sets of the bipartite tree action.
Tocompute conformal dimension we need metric estimates on boundaries. In thissection we estimate the distances and diameters of the limit sets of the vertexgroups appearing in the bipartite tree action of Proposition 2.5. We use K , K , . . . for the constants found in these estimates so that their use is clear later in thepaper.We think of the V T vertex spaces/groups as generalised edge spaces/groups,and indeed we do not need to consider edges any more, since every edge space isat finite Hausdorff distance from the adjacent V T vertex space. Nevertheless wekeep the notation so that v stands for a vertex in V T and e stands for a vertexin V T . So Lemma 2.1 becomes: Lemma 2.6. If G is a hyperbolic group with G (cid:121) T as in Proposition 2.5, with G acting geometrically on the model space Z , and X = ∂ ∞ Z with a visual metric,then every x ∈ X corresponds to exactly one of the following: • a point of ∂ ∞ T , with a unique x for each t ∈ ∂ ∞ T , or • a point of Λ e for some unique e ∈ V T , or • a point of Λ v for some unique v ∈ V T (but not in any Λ e ). As before, by quasiconvexity Λ v is a quasisymmetric image of g v ∂ ∞ G i v for each v ∈ V T . Likewise, for each e ∈ V T , Λ e is a quasisymmetric image of g e ∂ ∞ G k e ,that is, it is a pair of points in X .Fix corresponding basepoints v ∈ V T , o ∈ Z . For each v ∈ V T \ { v } , let e v ∈ V T be the last V T vertex on the geodesic from v to v . We have that Λ e v cuts X into at least two components [Bow98, Sec 1], while the interior of the openedge ( e v , v ) cuts T into exactly two components, one containing v and the othernot. Let Z ← e v be the component of Z \ Z ( e v ,v ) × { } containing o , and let Z e v → be the other component. We define U ← v := ∂ ∞ Z ← e v and U v → := ∂ ∞ Z e v → . Since Z ← e v and Z e v → are quasiconvex, these correspond to the closure of the limit setsof the corresponding components of T \ ( e v , v ). Note that U ← v ∩ U v → = Λ e v .We let U v → := X and leave U ← v and Λ e v undefined.We say that w ∈ V T is a descendant of v ∈ V T \ { v } if v separates w from v in T . We also say that all vertices of T are descendants of v . For v ∈ T , wedenote by T ( v ) the collection of v and all its descendants in V T .In all the following lemmas we assume as above that Z is a tree of spaces for agraph of groups decomposition of the group G like in Proposition 2.5.The following lemma implies that for any e (cid:54) = e (cid:48) ∈ V T , we have ∆(Λ e , Λ e (cid:48) ) ≥ /K , where ∆( U, V ) := d ( U, V )diam U ∧ diam V is the relative distance of U, V ⊂ X . Lemma 2.7.
There exists a constant K so that for e (cid:54) = e (cid:48) ∈ V T we have d (Λ e , Λ e (cid:48) ) (cid:23) K diam Λ e ∧ diam Λ e (cid:48) . Proof.
Pick loxodromic elements g, g (cid:48) so that g ±∞ = Λ e and ( g (cid:48) ) ±∞ = Λ e (cid:48) , and let (cid:96), (cid:96) (cid:48) be their translation lengths; as there are finitely many conjugation classes ofedge stabilizers, we may assume that (cid:96), (cid:96) (cid:48) are uniformly bounded away from 0 and ONFORMAL DIMENSION AND SPLITTINGS 15 oZ e Z e (cid:48) a bc o (cid:48) Z e Z e (cid:48) pL Figure 5.
Tree approximation for Lemma 2.7 ∞ , and that there are uniform bounds on the quasi-geodesic constants for n (cid:55)→ g n and n (cid:55)→ ( g (cid:48) ) n .Consider the tree approximation to geodesic axes for Λ e and Λ e (cid:48) as in Figure 5.Suppose, as in the left of the figure, the axes remain 2 δ Z -close for a large dis-tance L , where δ Z is the hyperbolicity constant for Z . Up to swapping g, g − thismeans that there is a point p so that for any i ≤ L/(cid:96) (cid:48) , the point g −(cid:98) i(cid:96) (cid:48) /(cid:96) (cid:99) ( g (cid:48) ) i p is uniformly close to p . Thus, by the uniform properness of G (cid:121) Z , there ex-ists L (cid:48) independent of e, e (cid:48) so that if L > L (cid:48) then there exist i (cid:54) = i so that g −(cid:98) i (cid:96) (cid:48) /(cid:96) (cid:99) ( g (cid:48) ) i = g −(cid:98) i (cid:96) (cid:48) /(cid:96) (cid:99) ( g (cid:48) ) i , hence (cid:104) g (cid:105) and (cid:104) g (cid:48) (cid:105) are commensurable, a contra-diction to the disjointness of Λ e , Λ e (cid:48) .Thus, up to a uniformly bounded error, the tree approximation of Λ e , Λ e (cid:48) mustlook like the right of Figure 5, for some c ≥
0. Up to swapping e, e (cid:48) , the positionof o in the tree approximation must look like that of o or o (cid:48) in the figure; supposethe former (the latter case is similar and easier), and label the other relevantdistances a, b , up to bounded error. One can compute that d (Λ e , Λ e (cid:48) ) (cid:16) e − (cid:15) ( a + b ) ,diam Λ e (cid:16) e − (cid:15)a , and diam Λ e (cid:48) (cid:16) e − (cid:15) ( a + b + c ) , so as a + b + c ≥ a + b we are done. (cid:3) The tree-of-spaces structure of Z implies the following bounds when we considerhow edge limit sets cut X . Lemma 2.8.
There exists a constant K so that for v, w ∈ V T \ { v } with w ∈ T ( v ) we have d ( U ← v , U w → ) (cid:16) K d (Λ e v , Λ e w ) . Proof.
Since Λ e v ⊂ U ← v and Λ e w ⊂ U w → , we have d ( U ← v , U w → ) ≤ d (Λ e v , Λ e w ). Inparticular, if Λ e v = Λ e w then d ( U ← v , U w → ) = 0 = d (Λ e v , Λ e w ), so we can assumethat Λ e v and Λ e w are disjoint.Suppose x ∈ U ← v and y ∈ U w → . By the quasiconvexity of Z e v , a geodesic from o to x must lie in the C -neighbourhood of Z ← e v . By the quasiconvexity of Z e w , ageodesic from o to y must consist of an initial segment of length d Z ( o, Z e w ) from o to a point within distance C of Z e w , then a tail which remains within distance C of Z e w → . If we let e = e v and e (cid:48) = e w as in the proof of Lemma 2.7 and considerFigure 5, this means that ( x | y ) ≤ a + b + C , and so d ( x, y ) (cid:23) e − (cid:15) ( a + b ) (cid:16) d (Λ e v , Λ e w )by the argument of Lemma 2.7. Thus d ( U ← v , U w → ) (cid:23) d (Λ e v , Λ e w ) and we aredone. (cid:3) Lemma 2.9.
There exists a constant K so that for any v ∈ V T \ { v } and p ∈ U v → , we have d ( p, U ← v ) (cid:16) K d ( p, Λ e v ) . Proof.
Take u ∈ V T so that v ∈ T ( u ) , d T ( u, v ) = 2. Then since Λ e v ⊂ Λ u ⊂ U ← v we have d ( p, U ← v ) ≤ d ( p, Λ e v ).Now suppose p / ∈ Λ e v . By the quasiconvexity of Z e v a geodesic γ from o to p travels from o to within C of a nearest point in Z e v to o , then travels within N C Z e v to a point q , then stays in Z e v → \ N C Z e v . Moreover, d ( p, Λ e v ) (cid:16) e − (cid:15)d Z ( o,q ) .Suppose we have y ∈ U ← v = ∂ ∞ Z ← e v . By the quasiconvexity of Z e v , a geodesicfrom o to y cannot stay close to γ past q , thus ( y | p ) ≤ d Z ( o, q ) + C (cid:48) . So d ( p, Λ e v ) (cid:16) e − (cid:15)d Z ( o,q ) (cid:22) d ( p, y ). Taking the infimum over all y ∈ U ← v , we conclude that d ( p, Λ e v ) (cid:22) d ( p, U ← v ). (cid:3) A vertex limit set Λ v , its parent edge limit set Λ e v and U v → , the part of X containing Λ v which Λ e v cuts out, all have comparable diameters. Lemma 2.10.
There exists a constant K so that for v ∈ V T \ { v } , we have diam Λ e v ≤ diam Λ v ≤ diam U v → ≤ K diam Λ e v . Proof.
As Λ e v ⊂ Λ v ⊂ U v → , the first two inequalities are trivial. Now as Λ e v istwo-ended, diam Λ e v (cid:16) e − (cid:15)d Z ( o,Z ev ) . By the quasiconvexity of Z e v any geodesic γ from o with a tail in Z e v → must satisfy d Z ( γ ( t ) , Z e v → ) ≤ C for all t ≥ d Z ( o, Z e v ).So by the definition of U v → , if x ∈ U v → then the geodesic ray from o to x musthave a tail in the C -neighbourhood of Z e v → also. Thus for two points x, y ∈ U v → ,the quasiconvexity of Z e v implies that the geodesic line from x to y will live in abounded neighbourhood of Z e v → , and hence ( x | y ) ≥ d Z ( o, Z e v ) − C thusdiam U v → = sup x,y ∈ U v → d ( x, y ) (cid:22) e − (cid:15)d Z ( o,Z e ) (cid:16) diam Λ e v . (cid:3) The (relative) diameter of limit sets reflect the configuration of the correspond-ing vertex spaces.
Lemma 2.11.
There exists a constant K so that for any v, w ∈ V T with w ∈ T ( v ) , we have diam Λ v (cid:16) K e − (cid:15)d Z ( o,Z v ) ≤ e − (cid:15)d T ( v ,v ) , and (2.12) diam Λ w diam Λ v (cid:16) K e − (cid:15)d Z ( p v ,Z w ) ≤ e − (cid:15)d T ( v,w ) , (2.13) where p v ∈ Z v is a closest point in Z v to o ∈ Z .Proof. The projection Z → T that collapses each vertex space to a point, andeach edge space to an edge is 1-Lipschitz, so the second inequalities are trivial.For the first inequality in (2.12), as Z v is quasi-isometric to the coset g v G i v and G i v is an infinite group, for any point p ∈ Z v there is a geodesic line γ so that d ( γ, p ) ≤ C and γ is in the C -neighborhood N C ( Z v ) of Z v . Suppose p v ∈ Z v isa closest point to o . As there is a geodesic line almost through p v which limitsto points in Λ v , we have diam Λ v (cid:23) e − (cid:15)d Z ( o,p v ) (cid:16) e − (cid:15)d Z ( o,Z v ) . On the other hand,for any distinct x, y ∈ Λ v if α is a geodesic line from x to y , by quasiconvexity α ⊂ N C ( Z v ), and so ( x | y ) (cid:38) d Z ( o, Z v ) − C and thus, taking the supremum overall x, y ∈ Λ v , diam Λ v (cid:22) e − (cid:15)d Z ( o,Z v ) .Let p v ∈ Z v and p w ∈ Z w be closest points to o in Z v and Z w , respectively. By(2.12) we have diam Λ w / diam Λ v (cid:16) e − (cid:15) ( d Z ( o,p w ) − d Z ( o,p v )) . By quasiconvexity and ONFORMAL DIMENSION AND SPLITTINGS 17 hyperbolicity, the geodesic from o to p w passes within distance C of p v , and p w iswithin C of a closest point to p v in Z w , thus | d Z ( p v , Z w ) − ( d Z ( o, p w ) − d Z ( o, p v )) | ≤ C, and the conclusion follows. (cid:3) Points in two different limit sets cannot be much closer to each other than theyare to their first common ancestor.
Lemma 2.14.
There exists a constant K so that if v ∈ V T , w, w (cid:48) ∈ T ( v ) with d T ( v, w ) = d T ( v, w (cid:48) ) = 2 and w (cid:54) = w (cid:48) , then for x ∈ U w → , d ( x, Λ v ) ≤ K d ( x, U w (cid:48) → ) .Proof. Suppose x ∈ U w → and y ∈ U w (cid:48) → . By quasiconvexity of edge and vertexspaces, there is a C so that a bi-infinite geodesic γ from x to y has an initial tail in N C ( Z e w → ), then a segment in N C ( Z v ), then a terminal tail in N C ( Z e w (cid:48) → ) (thesemay overlap). Let p be a closest point in γ to o ; necessarily p ∈ N C ( Z v ). Notethat e − (cid:15)d Z ( o,p ) (cid:16) d ( x, y ).Take a bi-infinite geodesic β in N C ( Z v ) passing within distance C of p . Since p is within C of a geodesic ray from o to x , either ( x | β ( −∞ )) ≥ d Z ( o, p ) − C (cid:48) or( x | β (+ ∞ )) ≥ d Z ( o, p ) − C (cid:48) . Without loss of generality, suppose the latter holds.Then d ( x, Λ v ) ≤ d ( x, β (+ ∞ )) (cid:22) e − (cid:15)d Z ( o,p ) (cid:16) d ( x, y ) . Taking the infimum of the right-hand side over all y ∈ U w (cid:48) → we get d ( x, Λ v ) (cid:22) d ( x, U w (cid:48) → ). (cid:3) Limit sets in the same orbit are, up to rescaling, uniformly bi-Lipschitz (as wedo not use the explicit constant later, we just call it C ). Lemma 2.15.
There exists C so that for any v ∈ T , the metric spaces (cid:26) gv Λ gv (cid:27) g ∈ G are all pairwise C -bi-Lipschitz.Proof. As there are finitely many vertex orbits it suffices to show the theorem fora fixed v ∈ T . We have that g Λ v = Λ gv for any g ∈ G by the equivariance of themap Z → T .By approximate self-similarity (Lemma 2.2) applied to a ball of radius diam Λ gv around a point of Λ gv , there exists h ∈ G so that the map (cid:18) Λ gv , gv d (cid:19) → ( h Λ gv , d )is bi-Lipschitz with uniform constant. Since diam h Λ gv = diam Λ hgv is then (cid:23) d Z ( o, Z hgv ) ≤ C for some constant C .Recall that g v o ∈ Z v , so gg v o ∈ gZ v = Z gv . Let g , . . . , g k ∈ G be chosen sothat any Z g (cid:48) v , g (cid:48) ∈ G , with d Z ( o, Z g (cid:48) v ) ≤ C has g (cid:48) v = g i v for some i ∈ { , . . . , k } .Moreover, we can choose g i so that g i g v o ∈ Z g i v is a closest point to o in the orbit Go ∩ Z g i v , and so d Z ( o, g i g v o ) ≤ C . Thus for any i, j ∈ { , . . . , k } , d Z ( o, g j g − i o ) ≤ d Z ( o, g j g v o ) + d Z ( g j g v o, g j g − i o ) ≤ C + d Z ( g i g v o, o ) ≤ C . Suppose for any two i, j ∈ { , . . . , k } we map Λ g i v to Λ g j v by h (cid:48) := g j g − i . Thenfor any two points x, y ∈ Λ g i v , we have d ( x, y ) (cid:16) e − (cid:15) ( x | y ) o = e − (cid:15) ( h (cid:48) x | h (cid:48) y ) h (cid:48) o , and d ( h (cid:48) x, h (cid:48) y ) (cid:16) e − (cid:15) ( h (cid:48) x | h (cid:48) y ) o . As | ( h (cid:48) x | h (cid:48) y ) o − ( h (cid:48) x | h (cid:48) y ) h (cid:48) o | ≤ d Z ( o, h (cid:48) o ) ≤ C , wethen have that the map h (cid:48) acts to send Λ g i v to Λ g j v in a uniformly bi-Lipschitzway.So in conclusion, by a uniformly bi-Lipschitz map one can send any of thespaces (cid:16) Λ gv , gv d (cid:17) to one of a finite set of spaces Λ g v , . . . , Λ g k v where eachdiam Λ g i v (cid:16)
1, and these spaces are each pairwise bi-Lipschitz with uniform con-stants. (cid:3)
A metric space X is C -uniformly perfect if for any x ∈ X, r ∈ (0 , diam X ), wehave B ( x, r ) \ B ( x, r/C ) (cid:54) = ∅ . This property is preserved by quasisymmetric maps,up to changing the constant C (see [Hei01, Exercise 11.2]). For completeness, werecall that a homeomorphism f : X → X (cid:48) is quasisymmetric if there exists ahomeomorphism η : [0 , ∞ ) → [0 , ∞ ) so that for all x, y, z ∈ X , d ( x, y ) ≤ td ( x, z )implies that d ( f ( x ) , f ( y )) ≤ η ( t ) d ( f ( x ) , f ( z )) [TV80]. Lemma 2.16.
There exists C so that for any v ∈ V T , the metric space Λ v is C -uniformly perfect.Proof. Suppose H is an infinite hyperbolic group, and ∂ ∞ H is endowed with avisual metric with visual parameter (cid:15) .If ∂ ∞ H has at least 3 points there is an ideal hyperbolic triangle limiting todistinct points y , y , y ∈ ∂ ∞ H . For any x ∈ ∂ ∞ H and r ∈ (0 , diam ∂ ∞ H ] thereexists h ∈ H so that the action of h moves the quasi-centre of the ideal triangle toa point p on the geodesic from the basepoint o to x at distance ≈ − (cid:15) log r from o .Inspecting the tree approximation to o, x, hy , hy , hy , we see that for at least one i ∈ { , , } , ( x | hy i ) o is approximately d H ( o, p ), and so d ∂ ∞ H ( x, hy i ) (cid:16) r . Thissuffices to show ∂ ∞ H is uniformly perfect.For any v ∈ V T , since Λ v has more than two points then as the map ∂ ∞ g v G i v → Λ v is a quasisymmetry, ∂ ∞ G i v has more than two points and so is uniformlyperfect. The composition ∂ ∞ G i v → g v ∂ ∞ G i v → Λ v is a quasisymmetry, and soΛ v is uniformly perfect too. By Lemma 2.15, up to rescaling the spaces { Λ gv } are uniformly bi-Lipschitz, so { Λ gv } are uniformly uniformly perfect. As there areonly finitely many vertex orbits in T we are done. (cid:3) Conformal dimension and Combinatorial modulus
In this section we describe how conformal dimension can be calculated usingcombinatorial modulus by work of [BK13, Car13]. Using this we reduce Theo-rem 1.4 to a statement about such modulus, Theorem 3.4 below.First, a complete metric space X is Ahlfors ( Q -)regular if for some Q ≥ µ on X so that for all x ∈ X, r ∈ (0 , diam X ] we have µ ( B ( x, r )) (cid:16) r Q . In such a situation Q must equal the Hausdorff dimension of X ,and moreover µ must be comparable to the Hausdorff Q -measure on X .If Z is a Gromov hyperbolic space admitting a geometric action (that is, aproper and cocompact action by isometries) by a finitely generated group, thenthe boundary ∂ ∞ Z endowed with a visual metric is Ahlfors regular by work ofCoornaert [Coo93]. We work with the following variation on Pansu’s conformaldimension. ONFORMAL DIMENSION AND SPLITTINGS 19
Definition 3.1.
Let X be a metric space. Then the (Ahlfors regular) conformaldimension of X is the infimum of all Q such that X is quasisymmetric to anAhlfors Q -regular space. If G is a Gromov hyperbolic group then Confdim ∂ ∞ G is a well-defined invariantof G , and if a group H is quasi-isometric to G then Confdim ∂ ∞ H = Confdim ∂ ∞ G .The (Ahlfors regular) conformal dimension of a space which is approximatelyself-similar can be calculated using estimates on ‘combinatorial modulus’ [BK13,Car13], which we now go on to describe.We fix a large constant a > a ≥ i ∈ N ,let X i be a maximal a − i -separated set in X , and let S i = { B ( x, a − i ) } x ∈ X i be thecorresponding cover of X .For δ > δ be the collection of all paths in X of diameter ≥ δ .Let ρ n : S n → [0 , ∞ ) be a function (a “weight function”). We say that ρ n isΓ δ -admissible if for any γ ∈ Γ δ , we have (cid:96) ρ n ( γ ) := (cid:88) A ∈S n ,A ∩ γ (cid:54) = ∅ ρ n ( A ) ≥ . The S n -combinatorial p -modulus of Γ δ is defined byMod p (Γ δ , S n ) := inf ρ n Vol p ( ρ n ) , where Vol p ( ρ n ) := (cid:88) A ∈S n ρ n ( A ) p and where we infimise over all Γ δ -admissible ρ n : S n → [0 , ∞ ). The criticalexponent for the p -modulus is defined by p c ( δ ) := inf (cid:110) p > n →∞ Mod p (Γ δ , S n ) = 0 (cid:111) . Theorem 3.2 (Keith–Kleiner, Carrasco [Car13, Corollary 3.13]) . If G is a hy-perbolic group acting geometrically on an unbounded geodesic (hyperbolic) space Z , with boundary at infinity X = ∂ ∞ Z endowed with a visual metric d and p c ( δ ) defined as above, then there exists δ > so that for all < δ ≤ δ , Confdim ∂ ∞ G = Confdim X = p c ( δ ) . Proof.
Such an X equipped with a visual metric satisfies the hypotheses of [Car13,Corollary 3.13] by Lemmas 2.2 and 2.4. (cid:3) In order to estimate p c ( δ ), it actually suffices to show that Mod p (Γ δ , S n ) isbounded independently of n , provided the maximum value of ρ n goes to zero: Lemma 3.3 (Bourdon–Kleiner [BK13, Corollary 3.7(3)]) . For any p ≥ and δ ,for some S n , Γ δ as above, if there exists ρ n : S n → [0 , ∞ ) weights that are Γ δ -admissible, and (cid:107) ρ n (cid:107) ∞ → as n → ∞ , and sup n Vol p ( ρ n ) < ∞ , then p c ( δ ) ≤ p .Proof. For any (cid:15) > p + (cid:15) ( ρ n ) = (cid:88) A ∈S n ρ n ( A ) p + (cid:15) ≤ (cid:107) ρ n (cid:107) (cid:15) ∞ Vol p ( ρ n ) → n → ∞ , therefore p c ( δ ) ≤ p + (cid:15) ; as (cid:15) was arbitrary we are done. (cid:3) So for each p bigger than our intended upper bound, it will suffice to find δ ∈ (0 , δ ) and such a sequence of weight functions. Theorem 3.4.
Suppose G is as in the statement of Theorem 1.4 and X a vi-sual metric on the boundary of the model space arising from the tree of cylindersconstruction of Proposition 2.5, and δ > is fixed.Then for any p > ∨ max { Confdim ∂ ∞ G i } , there exists weight functions ρ n on S n so that each ρ n is Γ δ -admissible, lim n →∞ (cid:107) ρ n (cid:107) ∞ = 0 , and the sequence Vol p ( ρ n ) is bounded. This theorem will be proved in subsequent sections, as we now summarise.
Proof.
The weights are defined in Section 4, up to a choice of parameters δ (cid:48) , E , E and E . Theorem 5.1 shows that lim n →∞ (cid:107) ρ n (cid:107) ∞ = 0 and fixes the value of E . Admissibility is shown, for suitable (now fixed) parameters δ (cid:48) , E and E , byTheorem 6.1. The uniform bounds on Vol p ( ρ n ) are then shown by Theorem 7.4. (cid:3) Proof of Theorem 1.4.
The lower boundConfdim ∂ ∞ G ≥ ∨ max { Confdim ∂ ∞ G i } follows from the fact that G is not virtually free and that each vertex group G i isquasiconvex in G .For the upper bound, let δ be given by Theorem 3.2 for X = ∂ ∞ G , and fix δ ∈ (0 , δ ]. By Theorem 3.2, Lemma 3.3 and Theorem 3.4 we then haveConfdim ∂ ∞ G = p c ( δ ) ≤ ∨ max { Confdim ∂ ∞ G i } . (cid:3) Candidate weight function
Our goal in this section is, given a choice of p > max { Confdim ∂ ∞ G i } , todefine suitable weight functions as in Theorem 3.4. The idea is similar to that ofthe example in Subsection 1.3: to iteratively define weights that turn geometricsequences of scales into arithmetic. There are additional complications which wedescribe as they arise.We continue with the notation of Section 2, and T is the tree of cylinders ofProposition 2.5 with V T = V T (cid:116) V T . Let v ∈ V T be the fixed basepoint in T . Projections to T . We project S n onto T as follows: for A ∈ S n , define the treeprojection π ( A ) ∈ V T to be the closest vertex to v in the convex hullConv ( v ∈ T V : Λ v ∩ A (cid:54) = ∅ )of all vertices whose limit set intersect A . The relationship between A and Λ π ( A ) is indicated by the following: Lemma 4.1.
There exists K ≥ so that for A ∈ S n , diam Λ π ( A ) ≥ K a − n , andthe distance from the centre of A to Λ π ( a ) is at most K a − n .Proof. If π ( A ) = v the bounds are trivial, so assume otherwise. Since A is centredon a point p ∈ U π ( A ) → and does not meet Λ e π ( A ) , by Lemma 2.9 d ( p, U ← π ( A ) ) (cid:16) d ( p, Λ e π ( A ) ) (cid:23) a − n , thus U π ( A ) → = X \ U ← π ( A ) contains a ball centred on p ofradius (cid:16) a − n . So by Lemma 2.10 and the uniform perfectness of X , diam Λ π ( A ) (cid:23) diam U π ( A ) → ≥ a − n .If A ∩ Λ π ( A ) (cid:54) = ∅ , we are done for any K ≥
1. Otherwise A ∩ Λ π ( A ) = ∅ , but bythe definition of π ( A ), A must meet U w → and U w (cid:48) → for two distinct w, w (cid:48) ∈ T ( v )with d T ( v, w ) = d T ( v, w (cid:48) ) = 2. Therefore by Lemma 2.14, d ( A, Λ π ( A ) ) (cid:22) a − n . (cid:3) ONFORMAL DIMENSION AND SPLITTINGS 21
Given v, w ∈ V T , let [ v, w ] ⊂ V T be the unique simple path from v to w .Suppose A ∈ S n and [ v , π ( A )] consists of v , v , . . . , v m = π ( A ). If v = v i forsome i ∈ { , , . . . , m − } then let v → A = v i +1 ; if v = π ( A ) then let v → A = π ( A );and if v / ∈ [ v , π ( A )] let v → A be undefined.Let us also define for any δ > T δ := Conv ( { v } ∪ { v ∈ V T : diam Λ v > δ } ) , which is the convex hull of the finite set of vertices in T whose limit sets are large(see the first equality in (2.12)); such sets will be used in the definition below. Model spaces.
We are given a choice of p > max { Confdim ∂ ∞ G i } , and want todefine suitable weight functions as in Theorem 3.4. For v ∈ V T , fix Q v ∈ [Confdim ∂ ∞ G v , p ), with the choice uniform on each G -orbit.For v ∈ V T , let D v = diam Λ v . For each G -orbit Gv ⊂ V T , the collec-tion of rescaled spaces { D gv Λ gv } are all uniformly bi-Lipschitz to each other(Lemma 2.15). For each v ∈ T , we fix a Q v -regular space X v = ( X v , d v ) ofdiameter 1 in the conformal gauge of ∂ ∞ G v , and an η -quasisymmetry map h v :Λ v → X v . Again by Lemma 2.15, X v and h v may be chosen so that the maps h gv : D gv Λ gv → X gv have X gv independent of g and the different maps h gv differ-ing from each other only by a uniform bi-Lipschitz homeomorphism. (This lastcondition means that there exists C so that for any g, g (cid:48) ∈ G , there exists a C -bi-Lipschitz homeomorphism f : D gv Λ gv → D g (cid:48) v Λ g (cid:48) v so that h gv = h g (cid:48) v ◦ f .) Finally,the distortion function η may be chosen uniformly for all v , as dilations do notaffect distortion.As η is fixed and the spaces Λ v are uniformly perfect with constant independentof v (Lemma 2.16), we can find τ ∈ (0 ,
1] and λ ≥ h v : D v Λ v → X v are uniform ( τ, λ )-bi-H¨older maps by [TV80, Theorem 3.14], i.e. for all v ∈ V T and all x, y ∈ Λ v ,(4.3) 1 λ (cid:18) d ( x, y ) D v (cid:19) /τ ≤ d v ( h v ( x ) , h v ( y )) ≤ λ (cid:18) d ( x, y ) D v (cid:19) τ . When we push the cover S n forward by h v to X v , it is useful to know that theimages are contained in balls of radius smaller than a − m v / m v ; by(4.3) we can take(4.4) m v := (cid:98) τ ( n + log a D v ) − log a (2 λ ) (cid:99) ∨ . Definition of weight function.
For each n ∈ N , and a constant E found later, wedefine the weight function ρ n : S n → R + by(4.5) ρ n ( A ) := E a − n (cid:89) v ∈ V T ρ nv ( A ) . For v ∈ V T and A ∈ S n , Λ v and A can interact in three ways according towhether v / ∈ [ v , π ( A )] , v ∈ [ v , π ( A )) or v = π ( A ). In the first case, we don’t want ρ nv to influence ρ n ( A ) at all; in the latter two we need to define a subset of Λ v corresponding to the location of A in or near Λ v :(4.6) W v,A := ∅ if v / ∈ [ v , π ( A )] , Λ e v → A if v ∈ [ v , π ( A )) ,B A otherwise , where B A is a ball in Λ v → A = Λ v of radius a − n centred on a point at most K a − n from the centre of A ; such a ball exists by Lemma 4.1.How shall we define ρ nv ( A )? The first ingredient is to distort according to h v :we want the relative size of W v,A in Λ v to match the relative size of h v ( W v,A ) in X v , so we have a factor of diam h v ( W v,A ) / W v,A /D v .The second ingredient is to transform geometric to arithmetic scales: for i =1 , . . . , m v the annulus of points at distance [ a − ( i +1) , a − i ] from h v Λ e v should besent to an annulus of width m v , so sets at distance ∼ a − i from h v Λ e v should bestretched by ∼ m v a − i . But we don’t want to do this to large vertex limit sets, orif a set is too close to Λ e v , as either could interfere with showing admissibility. So,for W ⊂ Λ v we let(4.7) f v ( W ) := (cid:40) d v ( h v W, h v Λ e v ) ≤ a − m v or v ∈ T δ (cid:48) , and m v d v ( h v W, h v Λ e v ) otherwise , where T δ (cid:48) is the finite subtree defined by (4.2) for a suitable parameter δ (cid:48) ∈ (0 , diam Λ v ) determined later. Note that we only use e v in (4.7) when it is definedsince v / ∈ T δ (cid:48) implies v (cid:54) = v .Combining the two deformations leads us to define, for each v ∈ V T ,(4.8) ρ nv ( A ) := d ( A, U ← v ) ≤ E a − n or m v ≤ W v,A = ∅ , anddiam h v ( W v,A )diam W v,A E D v f v ( W v,A ) otherwise.Here E , E , along with E from (4.5), are constants we choose later.By Lemma 2.11 and (4.4), for a given n there are finitely many v with m v > ρ n is well-defined, given choices of the constants δ (cid:48) , E , E and E .5. Bounding the maximum value of ρ n Recall that the idea of ρ n is to send a geometric sequence of annuli of points in Λ v at distance [ a − ( i +1) , a − i ] for i = 1 , . . . , k (and suitable k ) to an arithmetic sequenceof annuli of points at distance [ i +12 k +1 , i k +1 ]. In particular, for some v / ∈ T δ (cid:48) butwith Λ v (cid:16)
1, the smallest annulus in Λ v has size (cid:16) a − n , so is covered by boundedlymany balls in S n . The ρ n value of these balls will be (cid:16) /n , giving a heuristicestimate (cid:107) ρ n (cid:107) ∞ (cid:23) /n . This is essentially the worst case, as we now show. Theorem 5.1.
For ρ n as in Section 4, for large enough E and for any E , E , δ (cid:48) , lim n →∞ (cid:107) ρ n (cid:107) ∞ = 0 .Proof. Given A ∈ S n , consider the path [ v , π ( A )] = { v , v , . . . , v k = π ( A ) } in V T . By (4.5), (4.6), (4.8),(5.2) ρ n ( A ) = E a − n k (cid:89) i =0 ρ nv i ( A ) . In the proof we track the dependence of constants on E , E , E , δ (cid:48) . Step 1:
Let t ≥ v t ∈ T δ (cid:48) . As T δ (cid:48) is finite, t ≤ C for someconstant C = C ( δ (cid:48) ). For i ≤ t , we have f v i ( W v i ,A ) = 1 and D v i (cid:16)
1. For i < t , ONFORMAL DIMENSION AND SPLITTINGS 23 diam W v i ,A (cid:16) h v i ( W v i ,A ) (cid:16)
1. So(5.3) t − (cid:89) i =0 ρ nv i ( A ) (cid:16) C ( δ (cid:48) ,E ) ρ nv t ( A ) (cid:16) C ( δ (cid:48) ,E ) diam h v t ( W v t ,A )diam W v t ,A . Step 2:
A useful fact is the following: by Lemma 2.10, for 0 ≤ i < k , as W v i ,A = Λ e ( vi ) → A = Λ e vi +1 we have(5.4) 1 ≤ D v i +1 diam W v i ,A ≤ K . Step 3:
Consider the definition of ρ nv in (4.8). Suppose for some i ∈ { , . . . , k } we have d ( A, U ← v i ) ≤ E a − n or m v i ≤
1, then let s be the minimal such i . If nosuch i exists, set s = k + 1. If s ≤ k then either(1) d ( A, U ← v s ) ≤ E a − n , and so ρ nv i ( A ) = 1 for all i ≥ s ,(2) m v s ≤ D v s (cid:16) a − n , and so s ≤ k ≤ s + C for some C by Lemmas 2.11and 4.1. For each i ≥ s we have m v i ≤ C , diam h v i ( W v i ,A ) (cid:16)
1, and D vi diam W vi,A (cid:16) a − n a − n = 1. If m v i ≤ ρ nv i ( A ) = 1, else m v i ∈ (1 , C ] thus f v i ( W v i ,A ) (cid:16) ρ nv i ( A ) (cid:16) C ( E ) k (cid:89) i = s ρ nv i ( A ) ≤ C = C ( E ) . Step 4:
Now for every t < i < s we claim that(5.6) diam h v i ( W v i ,A ) f v i ( W v i ,A ) ≤ Cm v i . First, if i < k then by Lemma 2.7 the relative distance of W v i ,A = Λ e vi +1 and Λ e vi is bounded below. If i = k < s then as d ( A, U ← v k ) > E a − n we have that therelative distance of W v k ,A = B A and Λ e vk is bounded below by the definition of B A ,provided we fix E := K + 2 say, by Lemma 4.1. Since uniformly quasisymmetricmaps uniformly distort relative distances (e.g. [BK02a, Lemma 3.2]),diam h v i ( W v i ,A ) m v i d v i ( h v i W v i ,A , h v i Λ e vi ) ≤ C ( E ) m v i . Second, if d v i ( h v i W v i ,A , h v i Λ e vi ) ≤ a − m vi then for i < k since the relative distanceof W v i ,A and Λ e vi is ≥ /C , so the relative distance of h v i W v i ,A and h v i Λ e vi is ≥ /C , but this last relative distance is also ≤ a − m vi / diam h v i W v i ,A , wethus have diam h v i W v i ,A (cid:22) a − m vi . If i = k , as W v i ,A is a ball of radius a − n ,diam h v i ( W v i ,A ) (cid:22) ( a − n /D v i ) τ (cid:16) a − m vi . So for i < k or i = k in this second casewe have diam h v i ( W v i ,A ) f v i ( W v i ,A ) = diam h v i ( W v i ,A ) ≤ Ca − m vi ≤ Cm v i . Step 5:
By (5.2), (5.3), (5.5), (5.6) we have ρ n ( A ) (cid:22) C ( E ,δ (cid:48) ,E ,E ) a − n · diam h v t ( W v t ,A )diam W v t ,A · s − (cid:89) i = t +1 CD v i m v i diam W v i ,A If s − < t + 1 this last product is vacuous. In this case by (4.3) ρ n ( A ) (cid:22) C ( δ (cid:48) ) a − n (diam W v t ,A ) − τ (cid:22) a − n a − n (1 − τ ) = a − τn (cid:22) n . So we may assume t + 1 ≤ s − t + 1 ≤ i ≤ s − CD v i / diam W v i − ,A is bounded.As the sequence m v i is roughly decreasing at least linearly in i (by (2.13)), for allbut boundedly many terms at the tail of the sequence i = t + 1 , . . . , s − CD v i / ( m v i diam W v i − ,A ) ≤ C /m v i ≤
1. Once m v i is small (but still ≥ D v i (cid:16) a − n and diam W v i − ,A (cid:16) a − n also, so CD v i / ( m v i diam W v i − ,A ) (cid:22)
1. Takentogether, applying these bounds for i = t + 2 , . . . , s −
1, we have ρ n ( A ) (cid:22) a − n · diam h v t ( W v t ,A )diam W v t ,A · D v t +1 m v t +1 · W v s − ,A (cid:22) diam h v t W v t ,A m v t +1 by (5.4) and diam W v s − ,A (cid:23) a − n .If m v t +1 ≥ τ n/
2, then ρ n ( A ) (cid:22) n . Otherwise m v t +1 < τ n/ D v t (cid:16) h v t W v t ,A (cid:22) D τv t +1 (cid:16) a − τn + m vt +1 ≤ a − τn/ and ρ n ( A ) (cid:22) a − τn/ (cid:22) n .As in either case ρ n ( A ) (cid:22) n , we are done. (cid:3) Admissibility
Our goal in this section is to show that for δ < δ there are suitable choicesof parameters δ (cid:48) , E , E making the weight ρ n : S n → R as defined as in (4.5)admissible for Γ δ . We now treat the parameter E as a fixed constant given bySection 5. Theorem 6.1.
For δ < δ fixed, we can find δ (cid:48) ∈ (0 , δ ] and E , E large enoughindependent of n so that ρ n defined as in Section 4 is Γ δ -admissible for all n . Recall from (4.2) that T δ = Conv ( { v } ∪ { v ∈ T : diam Λ v > δ } )is the convex hull of the finite set of vertices in T whose limit sets are large.Curves in Γ δ need not be embedded and can start and end at arbitrary pointsin X ; the following proposition finds a nice subcurve for any γ ∈ Γ δ . Proposition 6.2.
There exist δ (cid:48) ∈ (0 , δ ] so that:Given γ ∈ Γ δ , we can find an arc ˆ γ ∈ Γ δ (cid:48) so that (1) ˆ γ is contained in the image of γ . (2) ˆ γ is contained in U v → and has endpoints at least δ (cid:48) apart in Λ v , for some v ∈ T δ (cid:48) . Before proving this, in the following lemma we relate points in X with pointsin ¯ T , the compactification of T . For x ∈ X , let Π( x ) ⊂ ¯ T be the correspondingpoint(s) in ¯ T determined by Lemma 2.6: Π( x ) is either a unique point in ∂ ∞ T ,a closed ball of radius 1 around a unique e ∈ V T (with x ∈ Λ e ), or a unique v ∈ V T (with x ∈ Λ v ). ONFORMAL DIMENSION AND SPLITTINGS 25
Lemma 6.3.
For G (cid:121) T as in Proposition 2.5, and Π as above, if C ⊂ X isconnected, then Π( C ) := (cid:83) x ∈ C Π( x ) is connected.Proof. Suppose Π( C ) is disconnected. Then as Π( C ) ⊂ ¯ T is a union of a subsetof V T , radius-1 balls around vertices in V T , and points of ∂ ∞ T , then there is avertex e ∈ V T \ Π( C ) so that Π( C ) meets more than one component of ¯ T \ { e } .Since C ∩ Λ e = ∅ , this means that C meets at least two components of X \ Λ e ,and so C is not connected. (cid:3) Proof of Proposition 6.2.
First, find an arc, that is, an embedded path γ : [0 , → X in the image of γ with endpoints diam( γ ) apart.Let w (cid:48) ∈ V T be the closest point to v in Π( γ ) ⊂ ¯ T , following the notation ofLemma 6.3. We call w ∈ V T a child of w (cid:48) if d T ( v , w ) = d T ( v , w (cid:48) ) + 2.If γ meets Λ w (cid:48) in exactly one or two points, those point(s) lie in some Λ e ⊂ Λ w (cid:48) for some e ∈ V T adjacent to w (cid:48) with d T ( v , e ) = d T ( v , w (cid:48) ) + 1. The points ofΛ e split γ into two or three subarcs each living in some U w → for some child w of w (cid:48) . Necessarilly, at least one of these subarcs has endpoints δ/ γ besuch a subarc of γ , and let w ∈ V T be the child of w (cid:48) with γ ⊂ U w → .If γ meets Λ w (cid:48) in more than two points, let γ = γ and let w = w (cid:48) . In eithercase, γ meets Λ w in more than two points, has endpoints at least δ/ γ lies in U w → .If there is a path in ¯ T \ { w } from Π( γ (0)) to Π( γ (1)), then as γ is an arc,there is e ∈ V T and two (possibly equal) children w (cid:48) , w (cid:48)(cid:48) of w so that γ consistsof an initial subarc in U w (cid:48) → that joins γ (0) to a point of Λ e , a subarc ˆ γ in U w → joining the endpoints of Λ e , and a final subarc from the other point of Λ e to γ (1)in U w (cid:48)(cid:48) → . Therefore by Lemma 2.10 δ ≤ d ( γ (0) , γ (1)) ≤ diam U w (cid:48) → + diam Λ e + diam U w (cid:48)(cid:48) → (cid:22) K +1 diam Λ e , so ˆ γ satisfies our desired property.So we now assume that w disconnects Π( γ (0)) from Π( γ (1)). This includesthe case that w is in one of these sets; if it is in both, γ already has our property.Let t (resp. t ) be the first (resp. last) time γ meets Λ w . If t >
0, the subarc γ | [0 ,t ] lives in U w − → for some child w − of w . Let t − be the first time γ meetsΛ w − , and if t − > w − be the child of w − with γ | [0 ,t − ] ⊂ U w − → . Similarly,if t <
1, let w be the child of w with γ | [ t , ⊂ U w → , let t be the last time γ meets Λ w , and if t < w be the child of w with γ | [ t , ⊂ U w → .We claim that we can take δ (cid:48) = δ/ K K and our desired arc ˆ γ to be γ | [ t i ,t i +1 ] for i = − , w i and livein U w i → for i = − , , i = − , , (cid:15) i = d ( γ ( t i ) , γ ( t i +1 )),when defined. Certainly if (cid:15) > δ (cid:48) we can take ˆ γ = γ | [ t ,t ] , so assume (cid:15) ≤ δ (cid:48) .If w is defined, either (cid:15) > δ (cid:48) and we are done, or (cid:15) ≤ δ (cid:48) . If w is defined thenthe tail γ | [ t , has diameter ≤ diam U w → ≤ K diam Λ e w ≤ K K d (Λ e w , Λ e w ) ≤ K K (cid:15) ≤ δ/ w − is defined, either (cid:15) − > δ (cid:48) and we aredone, or (cid:15) − ≤ δ (cid:48) , and then if w − is defined the subarc γ | [0 ,t − ] has diameter ≤ δ/ w i exist for i = − , − , , γ have distance ≤ δ/
100 + (cid:15) − + (cid:15) + (cid:15) + δ/ < δ/
10, a contradiction.
So for δ (cid:48) = δ/ K K , we have found a subarc ˆ γ which has endpoints in someΛ v , v ∈ T δ (cid:48) , which are δ (cid:48) -separated, and ˆ γ ⊂ U v → . (cid:3) We will use the following observation about the relative positions of cut pairs.
Lemma 6.4.
There exists C so that if v, w ∈ V T with v ∈ [ v , w ] , v (cid:54) = v , d T ( v, w ) = 2 and Λ e w = { p + , p − } ⊂ Λ v , then C ≤ d ( h v p + , h v Λ e v ) d ( h v p − , h v Λ e v ) ≤ C. Proof.
By symmetry it suffices to prove that d ( h v p + , h v Λ e v ) (cid:22) d ( h v p − , h v Λ e v ).Choose (not necessarily distinct) q − , q + ∈ Λ e v so that we have d ( h v p + , h v Λ e v ) = d ( h v p + , h v q + ) and d ( h v p − , h v Λ e v ) = d ( h v p − , h v q − ).By Lemmas 2.11 and 2.10 diam Λ e w (cid:22) diam Λ e v so by Lemma 2.7(6.5) d ( p − , p + ) = diam Λ e w (cid:22) d (Λ e w , Λ e v ) . In particular, d ( p − , p + ) (cid:22) d ( p − , q − ), and so by quasisymmetry d ( h v p − , h v p + ) (cid:22) d ( h v p − , h v q − ).Thus d ( h v p + , h v Λ e v ) = d ( h v p + , h v q + ) ≤ d ( h v p + , h v p − ) + d ( h v p − , h v q + ) (cid:22) d ( h v p − , h v q − ) + d ( h v p − , h v q + ) . (6.6)If q − = q + we are done as d ( h v p − , h v q − ) = d ( h v p − , h v Λ e v ).Suppose q − (cid:54) = q + . Since d ( h v p + , h v q + ) ≤ d ( h v p + , h v q − ), by the quasisymmetryof h − v , d ( p + , q + ) (cid:22) d ( p + , q − ). Combining this with (6.5),diam Λ e v = d ( q + , q − ) ≤ d ( q + , p + ) + d ( p + , q − ) (cid:22) d ( p + , q − ) ≤ d ( p + , p − ) + d ( p − , q − ) (cid:22) d (Λ e w , Λ e v ) + d ( p − , q − ) ≤ d ( p − , q − ) , therefore d ( h v q + , h v q − ) (cid:22) d ( h v p − , h v q − ). By Lemma 2.10, d ( p − , q + ) ≤ diam Λ v (cid:22) d ( q + , q − ), so d ( h v p − , h v q + ) (cid:22) d ( h v q + , h v q − ). Therefore d ( h v p − , h v q + ) (cid:22) d ( h v q − , h v q + ) (cid:22) d ( h v p − , h v q − ) , and applying this to (6.6) we are done. (cid:3) Proposition 6.7.
There are choices of parameters E , E so that there exists J > so that for all v ∈ T , and any arc β joining Λ e v in U v → , we have (6.8) (cid:88) A ∈S n : A ∩ β (cid:54) = ∅ (cid:89) w ∈ T ( v ) ρ nw ( A ) ≥ J a n diam Λ e v , where we take diam Λ e v := 1 . Moreover, if β is an arc in some U v → , v ∈ T δ (cid:48) withendpoints in Λ v that are δ (cid:48) -separated, then (6.9) (cid:96) ρ n ( β ) ≥ . Proof.
We prove that (6.8) holds in stages. Before we begin, we summarisethe dependence of constants chosen in the proof. All constants, in particular C , . . . , C ≥
1, depend on the data of our space and the constants K , . . . , K ≥ k , k ∈ N with k := (cid:100) log a (2 aλ ( K ∨ δ (cid:48) ) /τ ) (cid:101) and k := (cid:100) log a (6 K K ) (cid:101) .We choose j ∈ N based on Lemma 6.4. We introduce a parameter E which ischosen large enough depending on j , k , k , and set J := 1 /E . We find a constant ONFORMAL DIMENSION AND SPLITTINGS 27 C ∗ = C ∗ ( J ). The parameter E is chosen large enough depending on j , k , C ∗ (and C , C ). Finally we find C ∗ = C ∗ ( δ (cid:48) ) and set E := 1 / ( C ∗ J δ (cid:48) ) = E /C ∗ δ (cid:48) . Step 1 : Suppose v is a vertex with diam Λ v ≤ E a − n for a choice of E ≥ E below.(The important case is when v is the child of some ˆ v with diam Λ ˆ v ≥ E a − n .) Thusdiam Λ e v ≤ E a − n , so a n diam Λ e v ≤ E .In the left hand side of (6.8), for any A meeting Λ e v (as β does), we have that d ( A, U ← v ) = 0 so ρ nv ( A ) = 1. Also, for all w ∈ T ( v ) \ { v } , π ( A ) belongs to [ v , v ]so w / ∈ [ v , π ( A )], thus W w,A = ∅ and ρ nw ( A ) = 1 also. Therefore, the left-handside is ≥
1, and so (6.8) holds for J = 1 /E . Step 2:
Suppose v has diam Λ v ∈ [ E a − n , δ (cid:48) ) and all children of v satisfy (6.8)with J . Note that v (cid:54) = v .The idea is that by requiring E large enough, J doesn’t get worse in ourestimate for (6.8).If A ∈ S n meets β ⊂ U v → and has W v,A = ∅ , then as v / ∈ [ v , π ( A )], A mustalso meet U ← v . If d ( A, Λ e v ) ≥ a − n ≥ A then for any p ∈ A , and using K ≥ d ( A, Λ e v ) (cid:16) d ( p, Λ e v ) (cid:16) K d ( p, U ← v ) ≤ diam A ≤ a − n . Thus we conclude that(6.10) d ( A, Λ e v ) ≤ K a − n . The path β joins the endpoints of Λ e v , travelling through Λ v with subarcspassing through U w → for various children w of v .Let k , k be constants chosen as above, then add the condition m v > k + k to E , so that 0 ≤ k ≤ m v − k ≤ m v .For k ∈ { k , . . . , m v − k } , consider the points of X v = h v Λ v at distance( a − k , a − k +1 ] from h v Λ e v , and call this set Y k . By Lemma 6.4, there exists j sothat if a child w of v has Λ e w = { p + , p − } , and p + ∈ Y k , p − ∈ Y l , then | k − l | ≤ j .The pair Λ e v has diam Λ e v ≥ K diam Λ v by Lemma 2.10, so diam h v Λ e v ≥ λ − K − /τ by (4.3). Note that a − k +1 ≤ λ − K − /τ by the choice of k , so each Y k , k ≥ k , consists of two disjoint balls centred on h v Λ e v .Let M := (cid:98) ( m v − k − k − j − / (2 j + 1) (cid:99) ; add the condition that E islarge enough so that M ≥ i ∈ , . . . , M , consider k = k ( i ) := (2 j + 1) i + j + 1 + k ∈ { k + ( j + 1) , . . . , m v − k − ( j + 1) } , and the collection B i of subarcs of β that are either (i) in Λ v with h v -image in Y k ,or (ii) join Λ e w in some U w → with w a child of v and h v Λ e w meets or jumps over Y k . Here “jumps over” means that one end point lies in (cid:83) s>k Y s and the otherin (cid:83) s For k ≥ log a (6 K K ) , for any i = 0 , . . . , M and any A ∈ S n which intersects some arc in B i , we have that π ( A ) ∈ T ( v ) . Proof. Case 1, A ∩ Λ v (cid:54) = ∅ : By the definitions of B i , (4.4) and k , d ( h v ( A ∩ Λ v ) , h v Λ e v ) ≥ a − k − diam h v ( A ∩ Λ v ) ≥ a − k − a − m v ≥ a − m v (cid:16) a k + j − (cid:17) . Thus by (4.3) and (4.4), d ( A, Λ e v ) ≥ d ( A ∩ Λ v , Λ e v ) − diam A ≥ D v (cid:32) a − m v ( a k + j − ) λ (cid:33) /τ − a − n ≥ a − n (cid:16) a k + j − (cid:17) /τ − a − n > K a − n because k + j ≥ k ≥ log a (6 K K ) ≥ log a (6 K ). Thus by (6.10) we have W v,A (cid:54) = ∅ and as A ∩ Λ v (cid:54) = ∅ we have v = π ( a ). (Note that therefore W v,A = B A for a ball B A of radius a − n satisfying A ⊂ ( K + 1) B A , so if a collection of A ’scovers β (cid:48) , then the corresponding collection of ( K + 1) B A ’s also covers β (cid:48) .) Case 2: For some child w of v , A intersects a subarc β (cid:48) ⊂ U w → that meets orjumps over Y k . The endpoints of h v Λ e w lie in (cid:83) k + j s = k − j Y s , and both endpointshave the same closest point in h v Λ e v . It suffices to show that W v,A is either B A (as π ( A ) = v ) or Λ e w (as π ( A ) ∈ T ( w ) but π ( A ) (cid:54) = v ). This follows if we rule out π ( A ) ∈ [ v , v ).Similarly to Case 1, d ( h v Λ e w , h v Λ e v ) ≥ a − k − j ≥ a − m v a k so d (Λ e w , Λ e v ) ≥ a − n (cid:0) a k (cid:1) /τ . Since A ∩ β (cid:48) ⊂ U w → , Lemma 2.8 gives d ( A, Λ e v ) ≥ d ( U w → , U ← v ) − diam A ≥ K d (Λ e w , Λ e v ) − a − n ≥ a − n K − (2 a k ) /τ − a − n > K a − n , where the last inequality uses k ≥ log a (6 K K ), and so by (6.10) W v,A (cid:54) = ∅ . (cid:3) A jump β (cid:48) ∈ B i going through some U w → is large if there is some A ∈ S n with A ∩ β (cid:48) (cid:54) = ∅ and π ( A ) ∈ T ( w ). In this case, by Lemma 2.10 and Lemma 4.1,(6.12) diam Λ e w ≥ K diam U w → ≥ K diam Λ π ( A ) ≥ K K a − n . Suppose { β j } ⊂ B i are the large jumps in B i , going through U w j → for w j children of v . Consider the sets C i,j := { A ∈ S n : A ∩ β j (cid:54) = ∅} for each j . If A ∈ C i,j ∩ C i,j (cid:48) for j (cid:54) = j (cid:48) then A intersects both U w j → and U w j (cid:48) → ⊂ U ← w j so byLemma 2.9 d ( A, Λ e wj ) (cid:22) a − n . Therefore by (6.12), Lemma 2.7 and the doublingof X v we have that there exists C independent of v, i, j so that any A appears inat most C of these sets.For each choice of i (which fixes k ), we either have ‘many’ or ‘few’ large jumps. Case of many large jumps: Suppose (cid:80) j diam h v (Λ e wj ) ≥ a − k . Consider a given(large) jump h v Λ e wj . There is a constant C so that there are at most C many A ∈ S n with A ∩ β j (cid:54) = ∅ and π ( A ) = v ; for such A , (cid:81) u ∈ T ( w j ) ρ nu ( A ) = 1. By ONFORMAL DIMENSION AND SPLITTINGS 29 the Step 2 hypothesis (cid:80) A ∈S n : A ∩ β ∩ U wj → (cid:54) = ∅ (cid:81) u ∈ T ( w j ) ρ nu ( A ) ≥ J a n diam Λ e wj . So ifdiam Λ e wj ≥ (2 C /J ) a − n then we have (cid:88) A ∈S n : π ( A ) (cid:54) = v,A ∩ β ∩ U wj → (cid:54) = ∅ (cid:89) u ∈ T ( w j ) ρ nu ( A ) ≥ J a n diam Λ e wj . Moreover, when π ( A ) (cid:54) = v in this sum W v,A = Λ e wj so (cid:88) A ∈S n : A ∩ β ∩ U wj → (cid:54) = ∅ (cid:89) u ∈ T ( v ) ρ nu ( A ) = (cid:88) A ∈S n : A ∩ β ∩ U wj → (cid:54) = ∅ ρ nv ( A ) (cid:89) u ∈ T ( w j ) ρ nu ( A ) ≥ (cid:88) A ∈S n : π ( A ) (cid:54) = v,A ∩ β ∩ U wj → (cid:54) = ∅ ρ nv ( A ) (cid:89) u ∈ T ( w j ) ρ nu ( A ) ≥ E D v diam h v (Λ e wj ) m v a − k + j diam Λ e wj (cid:88) A ∈S n : π ( A ) (cid:54) = v,A ∩ β ∩ U wj → (cid:54) = ∅ (cid:89) u ∈ T ( w j ) ρ nu ( A ) ≥ E D v m v a − k + j J a n diam h v (Λ e wj ) . If diam Λ e wj < (2 C /J ) a − n holds, then diam Λ e wj (cid:16) C ( J ) a − n . So for any A ∈ S n with A ∩ β ∩ U w j → (cid:54) = ∅ and π ( A ) = v (and so W v,A = B A with diam B A (cid:16) a − n ),by the uniform quasisymmetry of h v we havediam h v Λ e wj diam h v B A (cid:16) (cid:16) diam Λ e wj diam B A , thus, at the cost of a constant C ∗ = C ∗ ( J ), we can replace diam h v B A / diam B A in the relevant ρ nv ( A ) by diam h v Λ e wj / diam Λ e wj , and so induction again gives (cid:88) A ∈S n : A ∩ β ∩ U wj → (cid:54) = ∅ (cid:89) u ∈ T ( v ) ρ nu ( A ) = (cid:88) A ∈S n : A ∩ β ∩ U wj → (cid:54) = ∅ ρ nv ( A ) (cid:89) u ∈ T ( w j ) ρ nu ( A ) ≥ E D v diam h v (Λ e wj ) m v a − k + j C ∗ diam Λ e wj (cid:88) A ∈S n : A ∩ β ∩ U wj → (cid:54) = ∅ (cid:89) u ∈ T ( w j ) ρ nu ( A ) ≥ E D v C ∗ m v a − k + j J a n diam h v (Λ e wj ) . Summing over all large jumps { β j } ⊂ B i we have (cid:88) A ∈ (cid:83) j C i,j (cid:89) u ∈ T ( v ) ρ nu ( A ) ≥ C (cid:88) j (cid:88) A ∈C i,j (cid:89) u ∈ T ( v ) ρ nu ( A ) ≥ C (cid:88) j E D v ( C ∗ ∨ m v a − k + j J a n diam h v (Λ e wj ) ≥ E D v C ( C ∗ ∨ m v a − k + j J a n a − k E D v C ( C ∗ ∨ m v a j J a n . Case of few large jumps: We have that (cid:80) j diam h v (Λ e wj ) < a − k , but the arcsand jumps in B i must total at least a − k +1 − a − k ≥ a − k in diameter. Thus thearcs and small jumps (that is jumps β (cid:48) where any A ∈ S n with A ∩ β (cid:48) (cid:54) = ∅ has π ( A ) = v ) must have total diameter in h v Λ v at least a − k .Suppose we have a small jump β (cid:48) through some U w (cid:48) → , w (cid:48) a child of v . The A which cover β (cid:48) each have a ball B A centred at a point of Λ v at most K a − n from the centre of A , so β (cid:48) is in the ( K + 1) a − n neighbourhood of Λ v . Thereforeby Lemma 2.9 we have β (cid:48) is in the K ( K + 1) a − n neighbourhood of Λ e w (cid:48) , sodiam Λ e w (cid:48) ≤ K ( K +1) a − n , and thus by Lemma 2.10 diam U w (cid:48) → ≤ K K ( K +1) a − n . So there exists C ≥ ( K + 1) so that if A meets the small jump β (cid:48) ,then C B A covers the entire small jump including its endpoints. Here for a ball B = B ( x, r ) and C > 0, we set CB := B ( x, Cr ).In the image then, if there is a sequence of arcs and small jumps connectingpoints at least some distance L apart, then the sum of diam h v ( C B A ) for those A covering the corresponding arcs in B i must total at least L . So as we do not havemany large jumps, we must have that (cid:80) A ∈S n ( B i ,v ) diam h v ( C B A ) ≥ a − k , where S n ( B i , v ) is defined to be the set of all A ∈ S n so that π ( A ) = v and A ∩ β (cid:48) (cid:54) = ∅ for some β (cid:48) ∈ B i .By uniform quasisymmetry diam h v ( C B A ) (cid:16) C diam h v ( B A ). Putting it to-gether, (cid:88) A ∈S n ( B i ,v ) (cid:89) u ∈ T ( v ) ρ nu ( A ) = (cid:88) A ∈S n ( B i ,v ) ρ nv ( A ) ≥ (cid:88) A ∈S n ( B i ,v ) E D v diam h v ( B A ) m v a − k + j diam B A ≥ C (cid:88) A ∈S n ( B i ,v ) E D v diam h v ( C B A ) m v a − k + j a − n ≥ E D v a − k C m v a − k + j a − n = E D v C m v a j a n . As for each i there are either many large jumps or not, we have: (cid:88) A ∈S n : A ∩ β (cid:54) = ∅ (cid:89) w ∈ T ( v ) ρ nw ( A ) ≥ M (cid:88) i =0 E D v m v a j (cid:18) J C ( C ∗ ∨ ∧ C (cid:19) a n = E D v ( M + 1) m v a j (cid:18) J C ( C ∗ ∨ ∧ C (cid:19) a n ≥ J (cid:18) ( M + 1)4 a j ( C ( C ∗ ∨ ∨ C ) m v (cid:19) E a n D v . By our earlier conditions on E , we have that M = (cid:98) ( m v − k − k − j − / (2 j +1) (cid:99) satisfies M ≥ 1. Thus ( M + 1) /m v is bounded away from zero, so we can anddo require that E is large enough depending on j , C , C ∗ , C so that the termin parentheses is at least 1 /E thus (6.8) holds for v with the same J . Note that D v = diam Λ v ≥ diam Λ e v . Step 3: Suppose v has diam Λ v ≥ δ (cid:48) and all children satisfy (6.8) for some J .The argument is identical to that of step 2 until we apply the definition of f v from (4.7), and as we don’t resize the annuli it suffices to consider the largestannulus Y k with i = 0 , k = j + 1 + k . We now indicate the slight differences. ONFORMAL DIMENSION AND SPLITTINGS 31 Case of many large jumps { β j } ⊂ B : If a given large jump β j has diam Λ e wj ≥ (2 C /J ) a − n then (cid:88) A ∈S n : A ∩ β ∩ U wj → (cid:54) = ∅ (cid:89) u ∈ T ( v ) ρ nu ( A ) ≥ (cid:88) A ∈S n : π ( A ) (cid:54) = vA ∩ β ∩ U wj → (cid:54) = ∅ ρ nv ( A ) (cid:89) u ∈ T ( w j ) ρ nu ( A ) ≥ E D v diam h v (Λ e wj )1 · diam Λ e wj (cid:88) A ∈S n : π ( A ) (cid:54) = vA ∩ β ∩ U wj → (cid:54) = ∅ (cid:89) u ∈ T ( w j ) ρ nu ( A ) ≥ E D v J a n diam h v (Λ e wj ) . While if diam Λ e wj < (2 C /J ) a − n then, for the same C ∗ = C ∗ ( J ) as before (cid:88) A ∈S n : A ∩ β ∩ U wj → (cid:54) = ∅ (cid:89) u ∈ T ( v ) ρ nu ( A ) ≥ E D v diam h v (Λ e wj )1 · C ∗ diam Λ e wj (cid:88) A ∈S n : A ∩ β ∩ U wj → (cid:54) = ∅ (cid:89) u ∈ T ( w j ) ρ nu ( A ) ≥ E D v C ∗ J a n diam h v (Λ e wj ) . Summing over all large jumps { β j } ⊂ B , as k = j + 1 + k we have (cid:88) A ∈ (cid:83) j C i,j (cid:89) u ∈ T ( v ) ρ nu ( A ) ≥ C (cid:88) j E D v ( C ∗ ∨ · J a n diam h v (Λ e wj ) ≥ E D v C ( C ∗ ∨ · J a n a − ( j +1+ k ) . Case of few large jumps { β j } ⊂ B : The argument is the same, giving the bound (cid:88) A ∈S n ( B ,v ) (cid:89) u ∈ T ( v ) ρ nu ( A ) ≥ (cid:88) A ∈S n ( B ,v ) E D v diam h v ( B A )1 · diam B A ≥ C (cid:88) A ∈S n ( B ,v ) E D v diam h v ( C B A )1 · diam B A ≥ E D v a − ( j +1+ k ) C diam B A ≥ E D v a − ( j +1+ k ) C a n . As B either has many large jumps or not, we have: (cid:88) A ∈S n : A ∩ β (cid:54) = ∅ (cid:89) w ∈ T ( v ) ρ nw ( A ) ≥ E D v a − ( j +1+ k ) (cid:18) J C ( C ∗ ∨ ∧ C (cid:19) a n ≥ J (cid:32) a − ( j +1+ k ) C ( C ∗ ∨ ∨ C ) (cid:33) E a n D v . Since D v = diam Λ v ≥ diam Λ e v , provided E is required to be large enoughdepending only on j , k , C , C ∗ , C we get (6.8) for v with the same value of J .As this point we fix the value of E . Conclusion: So we have shown that (6.8) for all v ∈ T . Suppose now that the curve β has endpoints in Λ v , v ∈ T δ (cid:48) , that are δ (cid:48) separated,but not necessarily agreeing with Λ e v . By the bi-H¨older estimates on h v for such v , this implies that the distance between the h v -images of the endpoints of β is ≥ λ − ( δ (cid:48) ) /τ > 0. Notice that in step 3 the location (if defined) of Λ e v was notrelevant, only that β crossed an annulus of width proportional to D v . So as wealready required k to satisfy a − k +1 ≤ λ − ( δ (cid:48) ) /τ , we can set i = 0, k = j +1+ k and let Y k be all points in h v Λ v at distance in ( a − k , a − k +1 ] from { h v β (0) , h v β (1) } ;note that the balls of radius a − k +1 centred on h v Λ v are disjoint. Step 3 then gives,with the same choice of E , (cid:88) A ∈S n : A ∩ β (cid:54) = ∅ (cid:89) w ∈ T ( v ) ρ nw ( A ) ≥ J a n D v . It remains to bound (cid:96) ρ n ( β ). Since β is in U v → , we have for any A ∈ S n with A ∩ β (cid:54) = ∅ that π ( A ) ∈ [ v , v ]. This means that if w ∈ V T has ρ nw ( A ) (cid:54) = 1then w ∈ T ( v ) or w ∈ [ v , v ]. Write { v , v , . . . , v m = v } ⊂ V T for the verticesof [ v , v ]; these are in T δ (cid:48) . For i = 0 , . . . , m − 1, we have diam W v i ,A (cid:16) δ (cid:48) h v i W v i ,A (cid:16) 1, and D v i (cid:16) so there is some constant C (cid:48) depending on δ (cid:48) so that ρ nv i ( A ) (cid:16) C (cid:48) E . Thus, using the trivial bound m ≤ | T δ (cid:48) | and setting C ∗ := ( C (cid:48) E ) | T δ (cid:48) | , (cid:96) ρ n ( β ) = E a − n (cid:88) A ∈ S n : A ∩ β (cid:54) = ∅ (cid:89) w ∈ V T ρ nw ( A )= E a − n (cid:88) A ∈ S n : A ∩ β (cid:54) = ∅ (cid:32) m − (cid:89) i =0 ρ nv i ( A ) (cid:33)(cid:32) (cid:89) w ∈ T ( v ) ρ nw ( A ) (cid:33) ≥ E a − n (cid:88) A ∈ S n : A ∩ β (cid:54) = ∅ (cid:0) C (cid:48) E (cid:1) | T δ (cid:48) | (cid:32) (cid:89) w ∈ T ( v ) ρ nw ( A ) (cid:33) ≥ E a − n C ∗ J a n D v ≥ E C ∗ J δ (cid:48) ≥ , thanks to our choice of E := 1 / ( C ∗ J δ (cid:48) ). (cid:3) Admissibility now follows. Proof of Theorem 6.1. For δ < δ , let δ (cid:48) > E , E be given by Proposition 6.7. By Proposition 6.2(2), any curve γ ∈ Γ δ has a subarc ˆ γ ∈ Γ δ (cid:48) , which Proposition 6.7 shows has (cid:96) ρ n (ˆ γ ) ≥ 1, so (cid:96) ρ n ( γ ) ≥ (cid:96) ρ n (ˆ γ ) ≥ 1. Therefore ρ n is Γ δ -admissible. (cid:3) Volume It remains to bound the volume of the weights ρ n . For v ∈ V T , recall that T ( v ) is the set of v and its descendants in V T . Let S n ( v ) := π − ( T ( v )), and S n ( v ) := π − ( v ). Note that for any v ∈ T we have the partition(7.1) S n ( v ) = S n ( v ) (cid:116) (cid:71) w child of v S n ( w ) . We define for v ∈ T ,(7.2) V n ( v ) := E p a − np (cid:88) A ∈S n ( v ) (cid:89) w ∈ T ( v ) ρ nw ( A ) p . ONFORMAL DIMENSION AND SPLITTINGS 33 Observe that the p -volume of ρ n is, by definition,Vol p ( ρ n ) = (cid:88) A ∈S n ρ n ( A ) p = E p a − np (cid:88) A ∈S n (cid:89) v ∈ V T ρ nv ( A ) p = V n ( v ) . (7.3)The goal of the section is the following, given fixed parameters δ (cid:48) , E , E , E . Theorem 7.4. We have V n ( v ) = Vol p ( ρ n ) bounded independently of n . We are going to set up an induction to bound the quantities V n ( v ). What isimportant here is the relative size of D v and a − n : for a given n and v ∈ V T ,our cover of U v → by balls of radius a − n is, if we rescale U v → by 1 /D v , like acover of D v U v → by balls of radius a − n /D v , and the corresponding p -volume isscaled by 1 /D pv . Let t v,n = (cid:98) n + log a D v (cid:99) , then the balls in the rescaled cover areapproximately of size a − t v,n .When t ≥ t for some fixed constant t ∈ Z set below, and t v,n = t for some n, v ,we are covering U v → by sets significantly smaller than U v → , and estimating V n ( v )is amenable to induction; the relevant quantity we try to bound is the following: Definition 7.5. For t ∈ Z , if t < t set ˆ V t = 1 , otherwise let ˆ V t be the supremumof V n ( v ) /D pv over all n , over all v ∈ V T \ T δ (cid:48) with t v,n = t . As an initial observation, for a given t there is an easy uniform bound on V n ( v ) /D pv with t v,n = t . Lemma 7.6. For each t ∈ Z fixed, ˆ V t < ∞ .Proof. The claim is trivial for t < t . We fix t ≥ t .Suppose for a given n that v ∈ T \ T δ (cid:48) has t v,n = t , i.e. (cid:98) n + log a D v (cid:99) = t , so a − np /D pv (cid:16) a − tp , and(7.7) V n ( v ) D pv (cid:16) a − tp (cid:88) A ∈S n (cid:89) w ∈ T ( v ) ρ nw ( A ) p . By definition, S n ( v ) consists of a collection of balls of radius a − n with centresseparated by a − n in U v → (not quite a cover since balls close to Λ e v will not beincluded). As X is Ahlfors regular, and by Lemma 2.10 diam U v → (cid:22) diam Λ v = D v , the number of such A included is bounded above by a constant depending on D v /a − n (cid:16) a t , i.e., depending only on t .For a given w ∈ T ( v ), the only way that ρ nw ( A ) (cid:54) = 1 is if d ( A, U ← w ) > E a − n and w ∈ [ v, π ( A )]. The first condition and Lemma 2.10 implies that diam Λ w (cid:23) a − n .The second condition and Lemma 2.11 implies that a − t (cid:16) a − n D v (cid:22) diam Λ w diam Λ v (cid:22) e − (cid:15)d T ( v,w ) , and so d T ( v, w ) is bounded by a constant depending on t . We have shown thatthe number of terms in the sum and in the product on the right-hand side of (7.7)are both bounded by a constant depending on t , and not on n .It remains to show that ρ nw ( A ) is bounded. This follows the argument inthe proof of Theorem 5.1. For those w that are in T ( w (cid:48) ) for some w (cid:48) with d ( A, U ← w (cid:48) ) ≤ E a − n or m w (cid:48) ≤ 1, then by cases (1) and (2) of Step 3 of the proofof Theorem 5.1, ρ nw ( A ) ≤ C ( E ). For all other w , we have d ( A, U ← w ) > E a − n and m w > 1, and by (5.6) in Step 4 of the proof of Theorem 5.1, we have diam( h w W w,A ) /f w ( W w,A ) (cid:22) /m w . Moreover, by Lemma 2.11 D w (cid:22) D v anddiam W w,A (cid:23) a − n , D w / diam W w,A (cid:22) D v /a − n (cid:16) a t . So ρ nw ( A ) = E D w diam W w,A · diam h w ( W w,A ) f w ( W w,A ) (cid:22) a t · m w (cid:22) a t , which is a constant bounded in terms of t .So we conclude that V n ( v ) /D pv is bounded by a constant depending on t , inde-pendent of n . (cid:3) We are going to bound V n ( v ) /D pv in two stages: a general inductive step usingˆ V t when v / ∈ T δ (cid:48) , then the finitely-many vertices of T δ (cid:48) will be dealt with using aweaker bound.7.1. Volume bounds for v / ∈ T δ (cid:48) . The goal of this subsection is to bound ˆ V t byinduction on t , and thus to bound V n ( v ) for v / ∈ T δ (cid:48) . In fact, we show more. Proposition 7.8. lim t →∞ ˆ V t = 0 . There are two kinds of contribution to V n ( v ), those coming from Λ v and thosecoming from U w → for some child w of v . Now for fixed n , a child w of v usuallyhas Λ w smaller than Λ v , and so t w,n < t v,n , and we can use an inductive boundto estimate the contribution of each child to V n ( v ); we have to treat carefully thefinitely many children that are exceptions.Note that if S n ( w ) (cid:54) = ∅ for w a child of v , then there exists A ∈ S n ( w ) so π ( A ) isa descendent of w , therefore by (2.13) and Lemma 4.1 diam Λ w ≥ K diam Λ π ( A ) ≥ K K a − n . Thus in the induction we are only interested in finitely many w : let C ( v ) = (cid:110) w child of v : diam Λ w ≥ K K a − n (cid:111) ⊂ V T. For such a w , t w,n = (cid:98) n + log a diam Λ w (cid:99) ≥ (cid:98)− log K K (cid:99) ; let t := (cid:98)− log K K (cid:99) . In our induction, the problem is that a child w of a given v ∈ T may have t w,n ≥ t v,n . However, by Lemma 2.11(2.13) for t ∆ := log K + 1 ≥ w ∈ T ( v ) then t w,n ≤ t v,n + t ∆ . To set up the induction, we group togetherthe finitely-many large descendents of v , where “large” depends on a parameter q ≤ t ∆ : let L ∗ ( v, q ) := { w ∈ T ( v ) \ { v } : ∃ u ∈ T ( w ) , t u,n ≥ t v,n − q }∪ ( T δ (cid:48) ∩ ( T ( v ) \ { v } )) , and L ( v, q ) := L ∗ ( v, q ) ∩ C ( v ) . Note that as v / ∈ T δ (cid:48) , T δ (cid:48) ∩ ( T ( v ) \{ v } ) = ∅ , but we will also use these definitionsof L ∗ ( v, q ) , L ( v, q ) later for more general v . There are uniform bounds |L ( v, q ) | ≤|L ∗ ( v, q ) | ≤ C ( t ∆ , δ (cid:48) ) for any v ∈ T : by (2.13) |L ∗ ( v, q ) | is bounded by the sum of | T δ (cid:48) | and the number of spaces Z w ⊂ Z which can meet a ball of a radius dependingon t ∆ and K . In the remainder of this subsection 7.1, v / ∈ T δ (cid:48) .Write L ( v ) := L ( v, , L ∗ ( v ) := L ∗ ( v, t ∈ Z defineˆ V Lemma 7.9. For t = t v,n = (cid:98) n + log a D v (cid:99) , if w ∈ C ( v ) \ L ( v ) then V n ( w ) /D pw ≤ ˆ V 1, i.e.(7.11) m v ≈ τ t v,n and m v (cid:16) τ t v,n . Our main technical bound in this subsection is the following. Proposition 7.12. There exists C depending on t ∆ and the data of our construc-tion so that for any v / ∈ T δ (cid:48) with t := t v,n ≥ t (cid:48) and any q ≤ t ∆ , we have V n ( v ) D pv ≤ Ct p − ˆ V < ( t − q ) + Ct p (cid:88) w ∈L ( v,q ) V n ( w ) D pw . Proof. We decompose V n ( v ) as follows:(7.13) V n ( v ) = E p a − np (cid:88) A ∈S n ( v ) (cid:89) w ∈ T ( v ) ρ nw ( A ) p = E p a − np (cid:88) A ∈S n ( v ) ρ nv ( A ) p (cid:124) (cid:123)(cid:122) (cid:125) (I) + E p a − np (cid:88) w ∈C ( v ) (cid:88) A ∈S n ( w ) ρ nv ( A ) p (cid:89) u ∈ T ( w ) ρ nu ( A ) p (cid:124) (cid:123)(cid:122) (cid:125) (II) . and we proceed to bound ( I ) and ( II ) in the following lemmas, whose proofs wedefer. (Recall that our parameters E , E , E , δ (cid:48) are now fixed constants.) Lemma 7.14. There exists C so that for any v / ∈ T δ (cid:48) with t = t v,n ≥ t (cid:48) we have ( I ) (cid:22) C a − np + D pv m pv . Lemma 7.15. There exists C depending on t ∆ so that for any v / ∈ T δ (cid:48) with t = t v,n ≥ t (cid:48) and q ≤ t ∆ we have ( II ) (cid:22) C a − np + D pv ˆ V < ( t − q ) m p − v + D pv (cid:88) w ∈L ( v,q ) m pv · V n ( w ) D pw . We now combine these bounds. By (7.13) and Lemmas 7.14 and 7.15: V n ( v ) D pv (cid:22) a − np D pv + 1 m pv + a − np D pv + ˆ V < ( t − q ) · m p − v + (cid:88) w ∈L ( v,q ) m pv · V n ( w ) D pw . As t = t v,n ≥ t (cid:48) we have (7.11), so we can write this as V n ( v ) D pv (cid:22) a − tp + 1 t p + 1 t p − ˆ V < ( t − q ) + 1 t p (cid:88) w ∈L ( v,q ) V n ( w ) D pw (cid:22) t p − ˆ V < ( t − q ) + 1 t p (cid:88) w ∈L ( v,q ) V n ( w ) D pw , where we use that ˆ V < ( t − q ) ≥ 1, so a − tp ≤ a − τtp (cid:22) t p (cid:22) t p − ˆ V < ( t − q ) . We have completed the proof of Proposition 7.12. (cid:3) Proposition 7.12 applies as follows to prove that lim t →∞ ˆ V t = 0. Proof of Proposition 7.8. Note that by Lemma 7.6, for t ∈ Z with t ≤ t < t (cid:48) wehave ˆ V t < ∞ , so we can restrict to values of t ≥ t (cid:48) where Proposition 7.12 applies.Our goal is to bound V n ( v ) /D pv in terms of ˆ V 0) and appears once.Since L ∗ ( v, 0) has a uniform finite bound on its size, after boundedly many stepsthis process terminates when there are no more vertices w (cid:48)(cid:48) with t w (cid:48)(cid:48) ,n ≥ t v,n . Atthe end we have V n ( v ) D pv ≤ C (cid:48) t p − ˆ V Bound (I).Proof of Lemma 7.14. We further split S n ( v ) = R ( v ) (cid:116) R ( v ) c where R ( v ) consistsof those A ∈ S n ( v ) with d ( A, U ← v ) > E a − n , and R ( v ) c = S n ( v ) \ R ( v ).( I ) = E p a − np (cid:88) A ∈R ( v ) c ρ nv ( A ) p + E p a − np (cid:88) A ∈R ( v ) ρ nv ( A ) p (cid:16) a − np |R ( v ) c | + a − np (cid:88) A ∈R ( v ) (cid:18) diam h v ( B A ) · D v diam B A · f v ( B A ) (cid:19) p (7.16) ONFORMAL DIMENSION AND SPLITTINGS 37 as for A ∈ S n ( v ), W v,A = B A . Lemma 2.9 gives that any A ∈ R ( v ) c is a distance ≤ K E a − n to Λ e v , so the doubling property of X gives(7.17) |R ( v ) c | ≤ C = C ( E , K ) . The value of f v ( B A ) depends on d v ( h v B A , h v Λ e v ), and so we write R ( v ) = R ( v, (cid:116) · · · (cid:116) R ( v, m v ) (cid:116) R ( v, m v + 1) where R ( v, j ) := { A ∈ R ( v ) : d v ( h v B A , h v Λ e v ) ∈ ( a − j , a − j +1 ] } , for 1 ≤ j ≤ m v and R ( v, m v + 1) = { A ∈ R ( v ) : d v ( h v B A , h v Λ e v ) ≤ a − m v } . So,pulling out for now the common factor a − np D pv , the second term of (7.16) is (cid:88) A ∈R ( v ) (cid:18) diam h v ( B A )diam B A · f v ( B A ) (cid:19) p = m v +1 (cid:88) j =1 (cid:88) A ∈R ( v,j ) (cid:18) diam h v ( B A )diam B A · f v ( B A ) (cid:19) p (cid:22) (cid:88) A ∈R ( v,m v +1) (diam h v ( B A )) Q v + p − Q v a − np · m v (cid:88) j =1 (cid:88) A ∈R ( v,j ) (diam h v ( B A )) Q v + p − Q v a − np m pv a − jp . (7.18)Recall that by (4.3), diam h v B A ≤ a − m v / 2. Since S n is a bounded multiplicitycover of X and each B A ⊂ ( K + 1) A the collection { B A : A ∈ S n ( v ) } has boundedmultiplicity, with constants depending only on X . Thus { h v B A : A ∈ S n ( v ) } is abounded multiplicity collection of quasi-balls, and so the Ahlfors Q v -regularity of X v gives, for 1 ≤ j ≤ m v + 1, (cid:88) A ∈R ( v,j ) (diam h v ( B A )) Q v diam h v ( B A ) p − Q v ≤ (cid:88) A ∈R ( v,j ) (diam h v ( B A )) Q v a − m v ( p − Q v ) (cid:22) a − ( j − Q v · a − m v ( p − Q v ) . (7.19)So when j = m v + 1 the right-hand side is a − pm v . By (7.19) the second term in(7.18) sums to (cid:22) m v (cid:88) j =1 a − ( j − Q v · a − m v ( p − Q v ) a − np m pv a − jp (cid:16) a − m v ( p − Q v )+ np m pv m v (cid:88) j =1 a j ( p − Q v ) (cid:22) a − m v ( p − Q v )+ np m pv · a m v ( p − Q v ) = a np m pv . (7.20)Combining (7.16),(7.17),(7.18),(7.19),(7.20) and a − pm v (cid:22) /m pv we have (cid:3) (7.21) ( I ) (cid:22) a − np + a − np D pv (cid:18) a − pm v a − np + a np m pv (cid:19) (cid:22) a − np + D pv · m pv . Bound (II).Proof of Lemma 7.15. Note that we are considering A ∈ S n ( w ) for some w ∈ C ( v ),i.e. π ( A ) equals w or a descendant of w , therefore W v,A = Λ e v → A = Λ e w in thisproof. Let us denote by ( IIa ) the contribution to ( II ) by w ∈ C ( v ) and A ∈ S n ( w )with d ( A, U ← v ) ≤ E a − n . For such w, A by Lemmas 2.7–2.10 we have a − n (cid:22) diam Λ w (cid:16) diam Λ e w (cid:22) min { diam Λ e w , diam Λ e v } (cid:22) d (Λ e w , Λ e v ) (cid:16) d ( U ← v , U w → ) ≤ d ( U ← v , A ) (cid:22) a − n , so all such w have diam Λ w (cid:16) a − n and also d ( A, Λ e v ) (cid:22) a − n by Lemma 2.9.By doubling and the separation property of Lemma 2.7 there are (cid:22) w .Moreover, ρ nv ( A ) = 1 and ρ nu ( A ) = 1 for all u ∈ T w , since U ← v ⊂ U ← u so d ( A, U ← u ) ≤ E a − n . So the total contribution to ( II ) by w, A as above with d ( A, U ← v ) ≤ E a − n is(7.22) ( IIa ) (cid:22) a − np . So in the remainder of this proof we only need to consider w ∈ C ( v ) and A ∈ S n ( w )with d ( A, U ← v ) > E a − n .In case ( I ) above we partitioned A ∈ S n ( v ) according to the distance values d v ( h v B A , h v Λ e v ); this time we partition the set of children of v according to boththeir distance d v ( h v Λ e w , h v Λ e v ) and their size diam h v Λ e w in the model space X v .We defined L ( v, q ) ⊂ C ( v ) to be, depending on q , those children with large descen-dents, and will consider their contribution in ( IIc ) below. For 1 ≤ j ≤ m v and1 ≤ k we partition the remaining children as follows: C ( v, q, j, k ) = (cid:110) w ∈ C ( v ) \ L ( v, q ) : d v ( h v Λ e w , h v Λ e v ) ∈ ( a − j , a − j +1 ] , and diam h v Λ e w ∈ ( a − k , a − k +1 ] (cid:111) , and we define C ( v, q, j, k ) similarly when j = m v + 1, replacing ( a − ( m v +1) , a − m v ]by (0 , a − m v ] in the appropriate place.There exists k ∆ so that if j ≥ k + k ∆ then C ( v, q, j, k ) = ∅ because Lemma 2.7gives a lower bound on the relative distance between Λ e w and Λ e v , and so thereis a uniform lower bound on the relative distance of h v Λ e w and h v Λ e v , since h v is η -quasisymmetric for uniform η [BK02a, Lemma 3.2].Also, as already remarked, C ( v ) is finite so only finitely many C ( v, q, j, k ) arenon-empty.Having this notation, we bound the terms of ( II ) not in ( IIa ) as follows: E p a − np (cid:88) w ∈C ( v ) (cid:88) A ∈S n ( w ) d ( U ← v ,A ) >E a − n ρ nv ( A ) p (cid:89) u ∈ T ( w ) ρ nu ( A ) p = E p a − np (cid:88) w ∈C ( v ) (cid:88) A ∈S n ( w ) d ( U ← v ,A ) >E a − n (cid:18) diam h v (Λ e w )diam Λ e w E D v f v (Λ e w ) (cid:19) p (cid:89) u ∈ T ( w ) ρ nu ( A ) p (cid:22) D pv (cid:32) ∞ (cid:88) k =1 ( m v +1) ∧ ( k + k ∆ ) (cid:88) j =1 (cid:88) w ∈C ( v,q,j,k ) a − kp V n ( w ) D pw f v (Λ e w ) p (cid:124) (cid:123)(cid:122) (cid:125) ( IIb ) + (cid:88) w ∈L ( v,q ) (diam h v (Λ e w )) p V n ( w ) D pw f v (Λ e w ) p (cid:124) (cid:123)(cid:122) (cid:125) ( IIc ) (cid:33) (7.23) ONFORMAL DIMENSION AND SPLITTINGS 39 We now use that v / ∈ T δ (cid:48) . Dropping for the moment the constant D pv , wedecompose ( IIb ) as(7.24) ( IIb ) ≤ ∞ (cid:88) k =1 m v ∧ ( k + k ∆ ) (cid:88) j =1 (cid:88) w ∈C ( v,q,j,k ) a − kp V n ( w ) D pw m pv a − jp + ∞ (cid:88) k =1 ∨ ( m v − k ∆ +1) (cid:88) w ∈C ( v,q,m v +1 ,k ) a − kp V n ( w ) D pw · , where we use that for w ∈ C ( v, q, j, k ), f v (Λ e w ) = 1 if j = m v + 1 and f v (Λ e w ) ≥ m v a j if j ≤ m v .In (7.24), each w considered is not in L ( v, q ), so t w,n < t v,n − q = t − q , so bythe definition of ˆ V < ( t − q ) , (7.24) is at most (cid:32) ∞ (cid:88) k =1 m v ∧ ( k + k ∆ ) (cid:88) j =1 (cid:88) w ∈C ( v,q,j,k ) a − kp m pv a − jp + ∞ (cid:88) k =1 ∨ ( m v − k ∆ +1) (cid:88) w ∈C ( v,q,m v +1 ,k ) a − kp (cid:33) ˆ V < ( t − q ) (7.25)Now by a volume estimate, for each j, k , (cid:88) w ∈C ( v,q,j,k ) a − kp = (cid:88) w ∈C ( v,q,j,k ) a − kQ v a − k ( p − Q v ) ≤ (cid:88) w ∈C ( v,q,j,k ) (diam h v (Λ e w )) Q v a − k ( p − Q v ) (cid:22) a − jQ v a − k ( p − Q v ) . (7.26)So (7.25) is at most ˆ V < ( t − q ) times ∞ (cid:88) k =1 m v ∧ ( k + k ∆ ) (cid:88) j =1 a − jQ v a − k ( p − Q v ) m pv a − jp + ∞ (cid:88) k =1 ∨ ( m v − k ∆ +1) a − ( m v +1) Q v a − k ( p − Q v ) (cid:22) m pv m v − k ∆ (cid:88) k =1 a k ( p − Q v ) a − k ( p − Q v ) + 1 m pv ∞ (cid:88) k = m v − k ∆ +1 a m v ( p − Q v ) a − k ( p − Q v ) + ∞ (cid:88) k = m v − k ∆ +1 a − m v Q v a − k ( p − Q v ) = 1 m pv m v − k ∆ (cid:88) k =1 m pv ∞ (cid:88) k = m v − k ∆ +1 a − ( k − m v )( p − Q v ) + ∞ (cid:88) k = m v − k ∆ +1 a − m v Q v − k ( p − Q v ) (cid:22) m p − v + 1 m pv + a − m v Q v − m v ( p − Q v ) (cid:22) m p − v . (7.27)where the implied constants depend on k ∆ , p − Q v and our other data.It remains to bound ( IIc ) from (7.23). We now show( IIc ) = (cid:88) w ∈L ( v,q ) (diam h v (Λ e w )) p f v (Λ e w ) p · V n ( w ) D pw (cid:22) (cid:88) w ∈L ( v,q ) m pv · V n ( w ) D pw . (7.28) This is true because if w ∈ L ( v, q ) we have t w ≥ t v − q − t ∆ ≥ t v − t ∆ , thusdiam Λ e w (cid:16) diam Λ e v , and so by (4.3) diam h v (Λ e w ) (cid:16) 1. Moreover d (Λ e w , Λ e v ) (cid:23) diam Λ e w by Lemma 2.7. By uniform relative distance distortion of quasisym-metric maps, we then get that d v ( h v Λ e w , h v Λ e v ) is bounded away from 0 for auniform constant. In the case d v ( h v Λ e w , h v Λ e v ) < a − m v then m v is bounded fromabove, so by (7.11) m v (cid:16) 1, thus f v (Λ e w ) = 1 (cid:16) m v . On the other hand if d v ( h v Λ e w , h v Λ e v ) ≥ a − m v then f v (Λ e w ) = m v d v ( h v Λ e w , h v Λ e v ) (cid:23) m v . In eithercase (7.28) holds.So in total, (7.22), (7.23), (7.24), (7.25), (7.27) and (7.28) give( II ) (cid:22) ( IIa ) + D pv (cid:0) ( IIb ) + ( IIc ) (cid:1) (cid:22) a − np + D pv ˆ V < ( t − q ) · m p − v + D pv (cid:88) w ∈L ( v,q ) m pv · V n ( w ) D pw . (cid:3) Volume bound for v ∈ T δ (cid:48) . For the boundedly many vertices in T δ (cid:48) , ourbound of Proposition 7.12 need not hold, however the following weaker bounddoes hold by a similar argument.Note that for all large enough n , for any v ∈ T δ (cid:48) we have t v,n ≥ t (cid:48) where t (cid:48) is the constant of (7.10), since t v,n ≈ n . So we assume from now on that for all v ∈ T δ (cid:48) we have τ t ≥ m v > m v ≈ τ t and m v (cid:16) τ t . Proposition 7.29. There exists C depending on t ∆ and the data of our construc-tion so that for any v ∈ T δ (cid:48) and any q ≤ t ∆ , we have V n ( v ) D pv ≤ C ˆ V < ( t − q ) + C (cid:88) w ∈L ( v,q ) V n ( w ) D pw Proof. We follow the proofs of Proposition 7.12 and Lemmas 7.14 and 7.15 inSubsection 7.1, but consider the case v ∈ T δ (cid:48) .Recall that by (7.13) we can write V n ( v ) as two sums, ( I ) where A ∈ S n ( v ),or ( II ) where π ( A ) is in S n ( w ) for some child w ∈ C ( v ). These are bounded asfollows; we defer the proofs until later. Lemma 7.30. There exists C so that for any v ∈ T δ (cid:48) we have ( I ) (cid:22) C a − np + D pv a − m v ( p − Q v ) . Lemma 7.31. There exists C depending on t ∆ so that for any v ∈ T δ (cid:48) and q ≤ t ∆ we have ( II ) (cid:22) C a − np + D pv ˆ V < ( t − q ) + D pv (cid:88) w ∈L ( v,q ) V n ( w ) D pw . We now combine (7.13), Lemmas 7.30 and 7.31 to find: V n ( v ) D pv (cid:22) a − np D pv + a − m v ( p − Q v ) + ˆ V < ( t − q ) + (cid:88) w ∈L ( v,q ) V n ( w ) D pw (cid:16) a − tp + a − τt ( p − Q v ) + ˆ V < ( t − q ) + (cid:88) w ∈L ( v,q ) V n ( w ) D pw (cid:22) ˆ V < ( t − q ) + (cid:88) w ∈L ( v,q ) V n ( w ) D pw , where we use that m v ≈ τ t and ˆ V < ( t − q ) ≥ 1. The proposition is proven. (cid:3) ONFORMAL DIMENSION AND SPLITTINGS 41 Bound (I).Proof of Lemma 7.30. We follow the notation and proof of Lemma 7.14. Theargument begins identically with (7.16) and (7.17).Instead of decomposing R ( v ) ⊂ S n ( v ) to find the bound (7.18), we instead usethe simpler fact that { h v B A : A ∈ S n ( v ) } is a bounded multiplicity collectionof quasi-balls in the Ahlfors Q v -regular space X v . Therefore, as f v ( B A ) = 1, wehave: (cid:88) A ∈R ( v ) (cid:18) diam h v ( B A )diam B A · f v ( B A ) (cid:19) p (cid:22) a np (cid:88) A ∈R ( v ) (diam h v B A ) Q v +( p − Q v ) ≤ a np a − m v ( p − Q v ) (cid:88) A ∈R ( v ) (diam h v B A ) Q v (cid:22) a np a − m v ( p − Q v ) . so instead of (7.21) we have( I ) (cid:22) a − np + a − np D pv (cid:16) a np a − m v ( p − Q v ) (cid:17) = a − np + D pv a − m v ( p − Q v ) . (cid:3) Bound (II).Proof of Lemma 7.31. We follow the argument and notation used in the proof ofLemma 7.15.As before, ( IIa ) satisfies the bound of (7.22). We write the remaining terms as D pv (cid:0) ( IIb ) + ( IIc ) (cid:1) as in (7.23). Since f v (Λ e w ) = 1, the bound (7.24) for ( IIb ) isreplaced by the following (again we drop for the moment the constant D pv ):(7.32) ( IIb ) = ∞ (cid:88) k =1 ( m v +1) ∧ ( k + k ∆ ) (cid:88) j =1 (cid:88) w ∈C ( v,q,j,k ) a − kp V n ( w ) D pw · . In (7.32), each w considered is not in L ( v, q ), so t w,n < t v,n − q = t − q , so bythe definition of ˆ V < ( t − q ) , (7.32) is at most(7.33) (cid:32) ∞ (cid:88) k =1 ( m v +1) ∧ ( k + k ∆ ) (cid:88) j =1 (cid:88) w ∈C ( v,q,j,k ) a − kp (cid:33) ˆ V < ( t − q ) Now the volume estimate (7.26) gives for each j, k , (cid:88) w ∈C ( v,q,j,k ) a − kp (cid:22) a − jQ v a − k ( p − Q v ) , so (7.33) is at most ∞ (cid:88) k =1 ( m v +1) ∧ ( k + k ∆ ) (cid:88) j =1 a − jQ v a − k ( p − Q v ) ˆ V < ( t − q ) ≤ ∞ (cid:88) k =1 ∞ (cid:88) j =1 a − jQ v a − k ( p − Q v ) ˆ V < ( t − q ) (cid:22) ˆ V < ( t − q ) . So in total, (7.22), (7.23), (7.32), (7.33), and the above give( II ) (cid:22) ( IIa ) + D pv (cid:0) ( IIb ) + ( IIc ) (cid:1) (cid:22) a − np + D pv ˆ V < ( t − q ) + D pv · ( IIc ) , where, as diam h v (Λ e w ) ≤ f v (Λ e w ) = 1,( IIc ) = (cid:88) w ∈L ( v,q ) (diam h v (Λ e w )) p f v (Λ e w ) p · V n ( w ) D pw ≤ (cid:88) w ∈L ( v,q ) V n ( w ) D pw . (cid:3) Uniform volume bounds. We can now complete the proof that V n ( v ) isbounded independently of n . Proof of Theorem 7.4. Our goal is to bound V n ( v ) (cid:16) V n ( v ) /D pv independentlyof n .First we apply Proposition 7.29 with v = v , t = t v ,n ≤ n, q = 0 (additionally, t ≈ n ) to bound V n ( v ) by V n ( v ) (cid:22) ˆ V In this section we characterise when the conformal dimension of a hyperbolicgraph of groups with elementary edge groups is attained. The key concept we useis porosity. Definition 8.1. A subset Y of a metric space X is porous if there exists c > sothat for any y ∈ Y and r ≤ diam( X ) there exists x ∈ X with B ( x, cr ) ⊂ B ( y, r ) \ Y . Under mild hypotheses, porosity is preserved by quasisymmetric homeomor-phisms. Lemma 8.2 (cf. [V¨ai87, Theorem 4.2]) . If X is a uniformly perfect metric space,and Y ⊂ X is porous, and f : X → X (cid:48) is a quasisymmetric homeomorphism, then f ( Y ) ⊂ X (cid:48) is porous. ONFORMAL DIMENSION AND SPLITTINGS 43 Proof. Given B (cid:48) = B ( y (cid:48) , r (cid:48) ) ⊂ X (cid:48) with y (cid:48) ∈ f ( Y ) and r (cid:48) ≤ diam X (cid:48) , since f − is quasisymmetric there exists r > B = B ( f − ( y (cid:48) ) , r ) satisfies B ⊂ f − ( B (cid:48) ) ⊂ λB , where λ ≥ f . Since Y isporous, there exists B ( x, cr ) ⊂ B \ Y . Now f ( B ( x, cr )) ⊂ f ( B ) ⊂ B (cid:48) , and byquasisymmetry there exists r (cid:48)(cid:48) > B ( f ( x ) , r (cid:48)(cid:48) ) ⊂ f ( B ( x, cr )) ⊂ B ( f ( x ) , λr (cid:48)(cid:48) ).Since B ( x, cr ) ⊂ B \ Y , B ( f ( x ) , r (cid:48)(cid:48) ) ⊂ B (cid:48) \ f ( Y ), so it remains to show that r (cid:48)(cid:48) /r (cid:48) ≥ c (cid:48) > c (cid:48) .In a uniformly perfect space, the radius of any ball is comparable to its diameter(indeed, this is an equivalent definition) up to some uniform constant C . So by[Hei01, Proposition 10.8], since B ⊂ f − ( B (cid:48) ) and diam B (cid:16) diam f − ( B (cid:48) ) wehave diam f ( B ) (cid:16) diam B (cid:48) (cid:16) r (cid:48) . Thus again by [Hei01, Proposition 10.8], writing η : [0 , ∞ ) → [0 , ∞ ) for the distortion function of f , r (cid:48)(cid:48) r (cid:48) (cid:16) diam f ( B ( x, cr ))diam f ( B ) ≥ η (cid:16) diam B diam B ( x,cr ) (cid:17) ≥ η ( C /c ) > . (cid:3) We will use the following criteria for non-attainment of Ahlfors regular confor-mal dimension, likely well-known to experts in the area. Proposition 8.3. Suppose there is a metric space X with a subset Y ⊂ X that isporous, so that the (Ahlfors regular) conformal dimensions of Y and X are equal(and finite). Then the conformal dimension of X is not attained.Proof. Suppose otherwise, and that f : X → X (cid:48) is a quasisymmetric map with X (cid:48) Ahlfors regular of dimension Confdim X . Since X (cid:48) is Ahlfors regular it is uniformlyperfect, and so is X = f − ( X (cid:48) ). Then f ( Y ) ⊂ X (cid:48) is porous by Lemma 8.2 above,so its Assouad dimension satisfies dim A f ( Y ) < dim A X (cid:48) = Confdim X [DS97,Lemma 5.8]. For any Q > dim A f ( Y ), f ( Y ) is quasisymmetric to an Ahlfors Q -regular space [Hei01, Theorem 14.16], so choosing Q ∈ (dim A f ( Y ) , Confdim X )we get that Confdim Y ≤ Q < Confdim X , a contradiction. (cid:3) A useful tool for identifying porous subsets is the following. Proposition 8.4. Suppose H is a quasiconvex subgroup of a hyperbolic group G .Then the following are equivalent: (1) the limit set Λ H is porous in ∂ ∞ G , (2) Λ H ⊂ ∂ ∞ G is a proper subset, and (3) H is infinite index in G .Proof. The implication (1) = ⇒ (2) is trivial. Likewise, (2) = ⇒ (3) is straightfor-ward: if [ G : H ] < ∞ then there is a bounded fundamental domain for the actionof H on a Cayley graph X for G , and so for some constant C , every point of X iswithin a distance C of H ⊂ X , and thus Λ H = ∂ ∞ X = ∂ ∞ G .It remains to show (3) = ⇒ (1). We fix a Cayley graph X for G . Since H acts freely on G , the quotient H \ X is a regular graph of bounded degree, with avertex for each right coset Hg . As [ G : H ] = ∞ , H \ X has infinite diameter, andso we can find a sequence of points g i ∈ G ⊂ X , i ∈ N , so that d ( H, g i ) → ∞ as i → ∞ . Suppose for each i that h i ∈ H ⊂ X is a closest point in H to g i . Let γ i : [0 , d ( h i , g i )] → X be a geodesic from h i to g i . By the choice of h i for each t ∈ [0 , d ( h i , g i )], d ( γ i ( t ) , H ) ≥ t . Let β i = h − i γ i , so that β i (0) = 1, and still foreach t in the domain of each β i , d ( H, β i ( t )) ≥ t . We apply Arzel`a–Ascoli to thesequence of maps ( β i ) to find a subsequence that converges uniformly on compact h yx x (cid:48) H y (cid:48) Figure 6. The configuration of the points x , x (cid:48) and y , with apotential location for y (cid:48) intervals to a map β : [0 , ∞ ) → X . This map β will be a geodesic ray, and willinherit the property that d ( β ( t ) , H ) ≥ t for all t ∈ [0 , ∞ ).Now to show porosity: fix a visual metric ρ on ∂ ∞ X = ∂ ∞ G , with visualparameter (cid:15) > C , so that ρ ( x, y ) (cid:16) C e − (cid:15) ( x | y ) . Suppose H is C -quasiconvex: any geodesic with endpoints on H lies in N C H ; this will also betrue for a geodesic ray from 1 ∈ H to a point of Λ H . Finally, write δ X for thehyperbolicity constant of X .We want to find c > y ∈ Λ H and r ≤ diam( ∂ ∞ X ), there exists x ∈ ∂ ∞ X with (i) for any y (cid:48) ∈ Λ H , ρ ( x, y (cid:48) ) ≥ cr , and (ii) for any x (cid:48) ∈ ∂ ∞ X with ρ ( x, x (cid:48) ) ≤ cr , ρ ( x (cid:48) , y ) ≤ r .We will set c = e − (cid:15) ( A + A ) , where A and A are parameters depending only (cid:15), C , C , δ X found below. Given y ∈ Λ H and r ≤ diam ∂ ∞ X , fix a geodesic ray α from 1 representing y . Consider the point of α at distance − (cid:15) log r + A from 1,and let h ∈ H be a point within C from that point. Let x ∈ ∂ ∞ X be the limitpoint of hβ .We show that (i) holds. For y (cid:48) ∈ Λ H , if ρ ( x, y (cid:48) ) < e − (cid:15) ( A + A ) r then ( x | y (cid:48) ) ≥ − (cid:15) log r + A + A − C for some C = C ( (cid:15), C ), so the geodesics from 1 to x and to y (cid:48) stay 2 δ X -close for all times up to this value. But this is a contradiction for large A since the geodesic from 1 to y (cid:48) lies in N C H , while at times t ≥ − (cid:15) log r + A ,the geodesic from 1 to x has distance at least t − ( − (cid:15) − log( r ) + A ) − C from H for C = C ( δ X , (cid:15), C , C ). See Figure 6.We show that (ii) holds. If ρ ( x, x (cid:48) ) ≤ cr = e − (cid:15) ( A + A ) then ( x | x (cid:48) ) ≥ − (cid:15) log r + A + A − C . If A is large enough, the tree approximation to 1, y , x and x (cid:48) mustlook like Figure 6. In particular, ( x (cid:48) | y ) equals − (cid:15) log r + A up to an additiveerror C . Thus ρ ( x (cid:48) , y ) ≤ C e − (cid:15) ( x (cid:48) | y ) ≤ C e − (cid:15)A e (cid:15)C r . Provided A is chosen largeenough depending on C , (cid:15) , C , we have ρ ( x (cid:48) , y ) ≤ r as desired. (cid:3) As an aside, this implies that hyperbolic groups which attain their conformal di-mension satisfy a kind of “co-Hopfian” property; compare the variations discussedin Kapovich–Lukyanenko [KL12] and Stark–Woodhouse [SW18]. (The second au-thor thanks Woodhouse for asking him this question.) Corollary 8.5. If G is a hyperbolic group, and ∂ ∞ G attains its (Ahlfors regu-lar) conformal dimension, then no finite index subgroup of G is isomorphic to aquasiconvex infinite index subgroup of G . ONFORMAL DIMENSION AND SPLITTINGS 45 Proof. Suppose H , H ≤ G are isomorphic (indeed, it suffices that they are quasi-isometric) with [ G : H ] < ∞ and [ G : H ] = ∞ . By Proposition 8.4, Λ H isporous in ∂ ∞ G . But Λ H and ∂ ∞ G are quasisymmetric, and hence each attainstheir conformal dimension, which contradicts Proposition 8.3. (cid:3) We return to our main goal, of characterising the attainment of conformal di-mension for a hyperbolic graph of groups with finite or 2-ended edge groups. Proof of Theorem 1.7. Suppose G is a hyperbolic group so that Confdim ∂ ∞ G isattained, and with a graph of groups decomposition over finite or 2-ended sub-groups.If Confdim ∂ ∞ G = 0 then G is virtually free by Stallings–Dunwoody, and asthe conformal dimension is attained G is 2-ended, see e.g. [MT10, Theorem 3.4.6].If Confdim ∂ ∞ G = 1 is attained, then G is virtually a cocompact Fuchsian groupby, e.g., a result of Bonk–Kleiner [BK02b, Theorem 1.1].We are left with the case that Confdim ∂ ∞ G > ∂ ∞ G i for some vertex group G i . Let T be theBass–Serre tree for the given graph of groups decomposition G . Each vertex of T is stabilized by a conjugate of a vertex group. If T has infinite diameter, then thereare infinitely many such vertices stabilized by conjugates of G i , each correspond-ing to a left coset of G i , so [ G : G i ] = ∞ , but this contradicts Proposition 8.3. So T has finite diameter.If there were any loops in G then T would have infinite diameter, so G mustbe a tree. Consider a leaf of G where the vertex group is G v and the adjacentedge group G e . Let x, y ∈ T be vertices stabilized by G i and G v respectively, and γ ⊂ T the simple path connecting them.If the injection i e : G e → G v has proper image, then the index [ G v : i ( G e )] ≥ y has degree ≥ 2, and there is an element g of G v ≤ G which fixes y butmoves the rest of γ . Since Confdim ∂ ∞ G i > 1, the edge groups adjacent to G i have infinite index in G i , so again there is an element g of G i ⊂ G which fixes x but moves the rest of γ . 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