Spaces and groups with conformal dimension greater than one
aa r X i v : . [ m a t h . M G ] A ug SPACES AND GROUPS WITH CONFORMALDIMENSION GREATER THAN ONE
JOHN M. MACKAY
Abstract.
We show that if a complete, doubling metric spaceis annularly linearly connected then its conformal dimension isgreater than one, quantitatively. As a consequence, we answera question of Bonk and Kleiner: if the boundary of a one-endedhyperbolic group has no local cut points, then its conformal di-mension is greater than one. Introduction
A standard quasi-symmetry invariant of a metric space (
X, d ) is its conformal dimension , introduced by Pansu [Pan89a]. It is defined asthe infimal Hausdorff dimension of all metric spaces quasi-symmetricto X , denoted here by dim C ( X ).Conformal dimension is a natural concept to consider since in somesense it measures the metric dimension of the best shape of X ; see[BK05a] for discussion and references for this kind of uniformizationproblem. A key application of the definition (and its original moti-vation) is in the study of the conformal structure of the boundary atinfinity of a negatively curved space.Besides the trivial bound given by the topological dimension of ametric space, the conformal dimension is often difficult to estimatefrom below. In this paper we give such a bound for an interesting classof metric spaces. Theorem 1.1.
Suppose ( X, d ) is a complete metric space which is dou-bling and annularly linearly connected. Then the conformal dimension dim C ( X ) is at least C > , where C depends only on the the constantsassociated to the two conditions above. Recall that a metric space is N -doubling if every ball can be coveredusing N balls of half the radius. The annularly linearly connectedcondition is a quantitative analogue of the topological conditions of Mathematics Subject Classification.
Primary 51F99; Secondary 20F67,30C65.This research was partially supported by NSF grant DMS-0701515. being locally connected and having no local cut points. This is madeprecise in Definition 3.2 and the subsequent discussion. For now, agood motivating example of a space satisfying our hypotheses is thestandard square Sierpi´nski carpet, denoted by S .The original motivation to study such spaces was given by a par-ticular application of Theorem 1.1. Each Gromov hyperbolic group G has an associated boundary at infinity ∂ ∞ G , a geometric object muchstudied in its relationship to the group structure of G (e.g. [Gro87,Bow98, Kle06]). The boundary carries a canonical family of metricsthat are pairwise quasi-symmetric, and so any quasi-symmetry invari-ant of metric spaces, such as conformal dimension, will give a quasi-isometry invariant of G .If the boundary of a hyperbolic group is connected and has no localcut points, for example if it is homeomorphic to the Sierpi´nski carpetor the Menger curve, then its self-similarity implies that it will satisfythe (a priori stronger) hypotheses of Theorem 1.1. Thus as a corollarywe answer a question of Bonk and Kleiner [BK05a, Problem 6.2]. Corollary 1.2.
Suppose G is a hyperbolic group whose boundary isnon-empty, connected and has no local cut points. Then the conformaldimension of ∂ ∞ G is strictly greater than one. These topological conditions on the boundary of a hyperbolic groupcorrespond to algebraic properties of the group itself. The boundary isnon-empty when the group is infinite. Using Stallings’ theorem on theends of a group, one sees that the boundary is connected if and only ifthe group does not split over a finite group [Sta68].More recently, work of Bowditch [Bow98, Theorem 6.2] shows thatif ∂ ∞ G is connected and not homeomorphic to S , then G splits overa virtually cyclic subgroup if and only if ∂ ∞ G has a local cut point.Using these results, we note a more algebraic version of Corollary 1.2. Corollary 1.3.
Suppose G is a one-ended hyperbolic group whose bound-ary has conformal dimension one. Then either the boundary of G ishomeomorphic to S (and hence G is virtually Fuchsian), or G splitsover a virtually cyclic subgroup. Outline of proof.
Let us return to the example of the standardSierpi´nski carpet, S . Since S has topological dimension equal to one,we need to rely on the metric structure of S to prove Theorem 1.1. Itis clear that S contains an isometrically embedded copy of C × [0 , C equals the standard one third Cantor set. By a lemma of ONFORMAL DIMENSION BOUNDS 3
Pansu, the conformal dimension of C × [0 ,
1] equals the Hausdorff di-mension of C plus one, and so we have that the conformal dimensionof S is greater than one; see, for example, [Pan89b, Example 4.3].In general, we do not have a product structure to exploit. Neverthe-less, we construct a family of arcs in our space X akin to the productof an interval and a regular Cantor set (of controlled dimension), andthen Pansu’s lemma completes the proof.Let us consider an example of extending a topological statement to aquantitative metric analogue. It is well known that a connected, locallyconnected, complete metric space X is arc-wise connected. Less wellknown is Tukia’s analogous metric result (Theorem 2.1): a linearlyconnected, doubling and complete metric space is connected by quasi-arcs. (See Section 2 for definitions.)If we now further assume that X has no local cut points – as in thesituation of Theorem 1.1 – then a topological argument shows that theproduct of a Cantor set and the unit interval embeds homeomorphicallyinto X . A weaker statement is that there exists a collection of arcs { J σ } in X such that, under the topology induced by the Hausdorff metric,the set { J σ } is a topological Cantor set.We will show a quantitatively controlled analogue of this weakerstatement. First, let ( M ( X ) , d H ) be the (complete) metric space con-sisting of all closed subsets of X with the Hausdorff metric d H . Foreach σ >
0, we shall denote by Z σ a standard Ahlfors regular Cantorset of Hausdorff dimension σ ; this is defined precisely in Section 3. Theorem 1.4.
For all L ≥ and N ≥ , there exist C ≥ , σ > and λ ′ ≥ such that if X is an L -annularly linearly connected, N -doubling,complete metric space of diameter at least one, then there exists a C -bi-Lipschitz embedding of Z σ into M ( X ) , where each point in the imageis a λ ′ -quasi-arc of diameter at least C . Moreover, on the image theHausdorff metric and minimum distance metric are comparable withconstant C . So, how do we create such a good collection of arcs? First, use thetopological properties of the space to split one arc into two arcs andapply Tukia’s theorem (Theorem 2.1) to straighten these arcs into uni-formly local quasi-arcs. Second, repeat this procedure in a controlledway by using the compactness properties of the quasi-arcs and spaces.This process gives four arcs, then eight, and so on, limiting to a col-lection of arcs indexed by a Cantor set. This process is described inSections 3 and 4.
J. M. MACKAY
In Section 4 we use Pansu’s lemma to complete the proof of Theo-rem 1.1. Corollary 1.2 follows from a short dynamical argument similarto one given by Bonk and Kleiner in [BK05b].As a final remark, we emphasize that the work here is to show theexistence of a uniform lower bound, greater than one, on the Haus-dorff dimension of any quasi-symmetrically equivalent metric. Pansugave examples of hyperbolic groups which do not have this property:the canonical family of (quasi-symmetrically equivalent) metrics on theboundary contains metrics whose Hausdorff dimension is arbitrarilyclose to, but not equal to, one. These groups are the fundamentalgroups of spaces obtained by gluing together two closed hyperbolicsurfaces along an embedded geodesic of equal length in each, corre-sponding to an amalgamation of the two surface groups along embed-ded cyclic subgroups. Of course, the boundaries of such groups containlocal cut points.For more discussion on conformal dimension, we refer the reader tothe Bonk and Kleiner [BK05a] and Kleiner [Kle06]. Note that theseauthors work with the Ahlfors regular conformal dimension; since thisinfimum is taken over a more restricted class of spaces, it is boundedbelow by the conformal dimension, and thus our result still applies.
Acknowledgments.
The author gratefully thanks Bruce Kleiner forall his help and advice. He also thanks the Department of Mathematicsat Yale University for its hospitality and Enrico Le Donne for commenson an earlier draft of this article.2.
Background
Quasi-arcs and arc straightening.
Basic analytic definitionsand results are contained in [Hei01]. Although conformal dimensionis defined using quasi-symmetric mappings, we will primarily use geo-metric arguments inside metric spaces.We will need some notation. A metric space (
X, d ) is said to be L -linearly connected for some L ≥ x, y ∈ X there exists acontinuum J ∋ x, y of diameter less than or equal to Ld ( x, y ). Thisis also known as the LLC(1) or BT (bounded turning) condition. Wecan actually assume that J is an arc, at the cost of increasing L by anarbitrarily small amount.As already mentioned, X is doubling if there exists a constant N suchthat every ball can be covered by at most N balls of half the radius.Note that a complete, doubling metric space is proper: closed balls arecompact. ONFORMAL DIMENSION BOUNDS 5
A key tool in creating the collection of arcs in Theorem 1.4 is a resultof Tukia that straightens arcs into local quasi-arcs. Before describingit we need some language to deal with embedded arcs. Denote thesub-arc of an arc A between x and y in A by A [ x, y ]. We say that anarc A in a doubling and complete metric space is an ǫ -local λ -quasi-arc if diam( A [ x, y ]) ≤ λd ( x, y ) for all x, y ∈ A such that d ( x, y ) ≤ ǫ . If thisholds for all ǫ >
0, then we say A is a λ -quasi-arc . The terminology isnatural since, by a result of Tukia and V¨ais¨al¨a [TV80], such an arc is(locally) the image of a quasi-symmetric embedding of the unit interval.One non-standard definition will be useful to us: we say that an arc B ǫ -follows an arc A if there exists a (not necessarily continuous) map p : B → A such that for all x, y ∈ B , B [ x, y ] is in the ǫ -neighborhoodof A [ p ( x ) , p ( y )]; in particular, p displaces points at most ǫ .We can now state Tukia’s theorem. Theorem 2.1 ([Tuk96, Theorem 1B]) . Suppose ( X, d ) is a L-linearlyconnected, N-doubling, complete metric space. For every arc A in X and every ǫ > , there is an arc J in the ǫ -neighborhood of A which ǫ -follows A , has the same endpoints as A , and is an αǫ -local λ -quasi-arc,where λ = λ ( L, N ) ≥ and α = α ( L, N ) > . Tukia’s original statement concerned subsets of R n . Bonk and Kleiner[BK05b, Proposition 3] used Assouad’s embedding theorem to trans-late it into this language. For a shorter proof, see [Mac08, Theorem1.1].As mentioned in the introduction, this theorem has the followingindependently interesting corollary: Corollary 2.2 (Tukia [Tuk96, Theorem 1A]) . Every pair of points ina L -linearly connected, N -doubling, complete metric space is connectedby a λ -quasi-arc, where λ = λ ( L, N ) ≥ . Hausdorff distance and Gromov-Hausdorff convergence.
We recall some standard definitions and results (for example, see [BBI01,Chapters 7,8]).Suppose (
X, d ) is a metric space. We define the distance between x ∈ X and U ⊂ X as d ( x, U ) = inf { d ( x, y ) : y ∈ U } . The r -neighborhood of U is the set N ( U, r ) = { x : d ( x, U ) < r } , where r ≥
0. The Hausdorff distance d H between U, V ⊂ X is d H ( U, V ) = inf { r ≥ U ⊂ N ( V, r ) , V ⊂ N ( U, r ) } . We say that a sequence of compact metric spaces { X i } , i ∈ N , con-verges to a metric space X in the Gromov-Hausdorff topology if there J. M. MACKAY exist f i : X → X i and ǫ i ≥ f i distorts distance by at mostan additive error of ǫ i , N ( f i ( X ) , ǫ i ) equals X i , and ǫ i →
0. (This isequivalent to the usual definition of convergence with respect to theGromov-Hausdorff metric.)If X is N -doubling and ǫ >
0, then X has a finite ǫ -net of cardinalityat most C ( N, ǫ ) < ∞ . Therefore, given any sequence of N -doublingspaces their geometry on scale ǫ can be modelled using uniformly finitesets. An Arzel`a-Ascoli argument gives the following result. For a proof,see [BBI01, Theorem 7.4.15]. Theorem 2.3.
Any sequence of N -doubling, complete metric spaces ofdiameter at most D has a subsequence that converges in the Gromov-Hausdorff topology to a complete metric space of diameter at most D . An analogous argument gives results when we consider configurationsof subsets inside X . As a simple example, consider a sequence of pairs { ( X i , A i ) } , where each A i is a closed subset of X i .We say that ( X i , A i ) converges to ( X, A ) in the Gromov-Hausdorfftopology, where A is a closed subset of X , if, as before, there exist f i : X → X i and ǫ i ≥ f i distorts distances by at most ǫ i , N ( f i ( X ) , ǫ i ) = X i , and ǫ i →
0. However, we now also require that d H ( f i ( A ) , A i ) ≤ ǫ i . At the cost of doubling ǫ i , we can assume that f i ( A ) ⊂ A i .A slightly modified version of the proof of Theorem 2.3 gives thefollowing: Theorem 2.4.
Suppose for each i ∈ N , X i is an N -doubling metricspace of diameter at most D , and A i is an arc in X i . Then thereis a subsequence of the configurations ( X i , A i ) that converges in theGromov-Hausdorff topology to a limit ( X, A ) .Moreover, if each A i is a λ -quasi-arc, then A will be a λ -quasi-arcalso; in particular, A is an arc. This last claim follows from an elementary argument using the defi-nitions of convergence and quasi-arcs.3.
Unzipping arcs
Consider a complete, locally connected metric space with no localcut points, that is, no connected open set is disconnected by removinga point. In such a space it is straightforward to “unzip” a given arc A into two disjoint arcs J and J lying in a specified neighborhoodof A . Repeating this procedure to get four arcs, then eight, and soon, it is possible, with some care, to get a limiting set homeomorphicto the product of a Cantor set and the interval. Such a limit set is ONFORMAL DIMENSION BOUNDS 7 useless for our purposes because there is no control on the minimumdistance between two unzipped arcs, and so no way to get a lowerbound on conformal dimension that is greater than one. We will usecompactness type arguments to overcome this problem.We begin by proving the topological unzipping result.
Lemma 3.1.
Given an arc A in a complete, locally connected metricspace with no local cut points, and ǫ > , it is possible to find twodisjoint arcs J and J in N ( A, ǫ ) such that the endpoints of J i are ǫ -close to the endpoints of A . Furthermore, the arcs J i ǫ -follow the arc A .Proof. Here, B ( x, r ) denotes the connected component of an open ball B ( x, r ) ⊂ X that contains its center x . As X is locally connected, B ( x, r ) is always open and connected, and, moreover, B ( x, r ) \ { x } isalso open and connected because x is not a local cut point. Any openand connected subset of X is arcwise connected.Let a and b be the initial and final points of A respectively (in afixed order given by the topology). We are going to define J and J inductively. There exists w ∈ B ( a, ǫ ) \ A ; otherwise, there would bea open set in X homeomorphic to an arc segment, violating the “nolocal cut point” condition. Now join w to a by an arc in B ( a, ǫ ). Stopthis arc at x , the first time it meets A , and call it J = J [ w, x ]. Set J = A [ a, x ]. (Perhaps x = a , but this is not a problem).Now we have two head segments for J and J meeting only at x ∈ A , and we want to unzip this configuration further along A . This ispossible since in B ( x, ǫ ) there is a tripod type configuration with twoincoming arcs J and J and one outgoing arc A [ x, b ]. As noted above, B ( x, ǫ ) \ { x } is arcwise connected, and so we can find an arc in this setthat joins some point in J (not x ) to a point in A [ x, b ] (not x ). Thearc may meet J , J and A [ x, b ] in many places but there must be somesub-arc A ′ joining some point in J or J to some point y in A withinterior disjoint from them all. (See Figure 1, where A ′ is emphasized.)Use A ′ to detour one of J and J around x to the new unzipping point y , and extend the other J i to y using A [ x, y ].What if this unzipping process approaches a limit before we are ǫ -close to the final point b in A ? This cannot happen. Suppose it is notpossible to unzip past z ∈ A . Since B ( z, ǫ ) \ { z } is arcwise connected,inside this set we can construct an arc A ′′ that detours around z , from z ∈ A to z ∈ A , where z < z < z in the order on A .Now by the limit point hypothesis, we can unzip J and J past z to x , where z < x < z . To continue the construction of J and J past z , find the arc given by following z to z along A ′′ , stopping if one of J. M. MACKAY
Figure 1.
Unzipping an arc J or J is met. If we reach z without intersecting J or J , as is thecase in Figure 2, then continue to follow A from z towards z . By theconstruction of J and J , this arc will meet J or J before reaching z .In either case, this arc can be used as a legitimate detour around x and z , contradicting the assumption on z . Thus it is possible to continueunzipping until x ∈ B ( b, ǫ ).It remains to find labellings f i : J i → A , for i = 1 ,
2. Define f i to be the identity on J i ∩ A . Each element v of J i \ A was created todetour around some point x ∈ A ; define f i ( v ) to equal x . This labellingcoarsely preserves order as desired. (cid:3) We would like to give a lower bound for the distance between thetwo split arcs. To do this we need a quantitative metric version ofbeing locally connected with no local cut points. Let A ( p, r, R ) be theannulus B ( p, R ) \ B ( p, r ). Definition 3.2.
We say a metric space X is ( L -)annularly linearlyconnected for some L ≥ p ∈ X , and x, y ∈ A ( p, r, r )for some r >
0, there exists an arc J joining x to y that lies in theannulus A ( p, rL , Lr ). Furthermore, we assume that X is connectedand complete.At the cost of replacing L by 8 L , we may assume that such a spaceis also L -linearly connected. ONFORMAL DIMENSION BOUNDS 9
Figure 2.
Avoiding a limit pointThis condition is stronger than the usual LLC (linearly locally con-nected) condition [HK98, Definition 3.12], and is mentioned in [HK98,Remark 3.19]. It is called LLC (linearly locally convex) in [BMS01,Section 2]; the authors of this paper use this condition in the contextof spaces that satisfy a Poincar´e inequality.The key feature of Definition 3.2 is that, unlike the usual LLC condi-tion, it preserved under Gromov-Hausdorff convergence. To be precise,if { X i } is a sequence of L -annularly linearly connected, uniformly dou-bling, complete metric spaces and X i → X ∞ in the Gromov-Hausdorfftopology, then X ∞ is L ′ -annularly linearly connected for any L ′ > L .(We need to increase L slightly to allow ourselves to connect by arcsrather than just continua.) Furthermore, annularly linearly connectedimplies that there are no local cut points.As a side remark, let us note that we do need a stronger condi-tion than no local cut points as a hypothesis for Theorem 1.1: it isstraightforward to modify the Sierpi´nski carpet construction to get adoubling, linearly connected, complete metric space with no local cutpoints whose Hausdorff dimension is one. Therefore, its conformal di-mension is also one.Now for the remainder of this section we will assume that L and N are fixed constants, and λ ≥ α ∈ (0 ,
1] are as given by Theorem 2.1.Consider the collection C of all λ -quasi-arcs A in any complete metricspace X that is L -annularly linearly connected and N -doubling, and whose endpoints a and b satisfy d ( a, b ) ∈ [ R , R ] for some R ≥
1. Fix ǫ >
0, and consider the supremum of possible separations of two arcssplit from A by the topological lemma above. Call this δ A ( δ A > Lemma 3.3.
There exists δ ⋆ = δ ⋆ ( λ, L, N, ǫ, R ) > such that for all A ∈ C , δ A > δ ⋆ .Proof. If not, then we can find a sequence of arcs A i ⊂ X i such that δ A i < i . Let a i and b i denote the endpoints of A i . We are onlyinterested in what happens inside the ball B i := B ( a i , L ( λR + ǫ )).As the sequence of configurations ( B i , A i , a i , b i ) is precompact in theGromov-Hausdorff topology, by an argument similar to Theorem 2.4we can take a subsequence converging to ( B ∞ , A ∞ , a ∞ , b ∞ ), where A ∞ is a λ -quasi-arc inside B ∞ with endpoints a ∞ and b ∞ .Convergence here means that there exist constants C i → f i : B ∞ → B i such that f i distorts distances by an additive error of atmost C i , and every point of B i is within C i of f i ( B ∞ ). Furthermore, f i ( A ∞ ) ⊂ A i , f i ( a ∞ ) = a i and f i ( b ∞ ) = b i .Since B ∞ will be L -annularly linearly connected (away from the edgeof the ball), it will have no local cut points in its interior. Consequently,we can split A ∞ into two arcs J and J using Lemma 3.1 inside an ǫ -neighborhood of A ∞ . These arcs are disjoint so they are separatedby some distance 0 < δ ′ ≤ ǫ . The remainder of the proof consistsof showing that this contradicts the assumption on A i ⊂ B i for somelarge i .For sufficiently large i , C i ≤ δ ′ L because C i → i → ∞ . For j = 1 ,
2, the arc J j in B ∞ contains a discrete path D j with C i -sizedjumps that corresponds to a discrete path D ′ j = f i ( D j ) in X i with2 C i ≤ δ ′ L jumps. The L -linearly connected condition can then be usedto join each D ′ j up into a continuous arc J ′ j .To be precise, if D ′ j = { p , . . . , p M } , join p to p by an arc J ′ j ofdiameter at most 2 C i L ≤ δ ′ . Assume that, at a stage k , we have an arc J ′ j from p to p k . There is an arc I of diameter at most δ ′ joining p k +1 to p k . We extend J ′ j to p k +1 by following I from p k +1 to p k , stopping at x , the first time it meets J ′ j , and gluing together J ′ j [ p , x ] and I [ x, p k +1 ]to make a new arc J ′ j , and we repeat this until k = M . Define a map h j : J ′ j → D ′ j that sends each of the points added at stage k to thepoint p k . Note that for all x, y ∈ J ′ j , J ′ j [ x, y ] ⊂ N ( D ′ j [ h j ( x ) , h j ( y )] , δ ′ );in a coarse sense, J ′ j δ ′ -follows D ′ j .By construction, J ′ and J ′ are δ ′ -separated and ǫ -close to A i , but toget a contradiction we need them to ǫ -follow A i . ONFORMAL DIMENSION BOUNDS 11
Since A ∞ and A i are both λ -quasi-arcs, Lemma 3.4 below impliesthat for all x , y ∈ A ∞ , f i ( A ∞ [ x, y ]) is contained in the ((2 C i λ + C i ) λ + C i )-neighborhood of A i [ f i ( x ) , f i ( y )]. For each j , we can lift the map h j : J ′ j → D ′ j to a map h ′ j : J ′ j → D j ⊂ B ∞ . By Lemma 3.1, D jǫ -follows A ∞ , so further compose with the associated map D j → A ∞ .Finally, compose with f i : A ∞ → A i .The composed maps J ′ j → D j → A ∞ → A i , for each j , show thateach J ′ j follows A i with constant (cid:0) δ ′ + C i + ǫ + (2 C i λ + C i ) λ + C i (cid:1) .This is smaller than ǫ for sufficiently large i because C i → i → ∞ .We have contradicted our initial assumption, so the proof is complete. (cid:3) We used the following lemma in the proof:
Lemma 3.4. If A and A ′ are λ -quasi-arcs, and f : A → A ′ is a mapdistorting distances by at most C , then for all x and y in A , f ( A [ x, y ]) ⊂ N ( A ′ [ f ( x ) , f ( y )] , (2 Cλ + C ) λ + C ) . Proof.
Let x = p < p < · · · < p n = y be a chain of points in A sothat the diameter of A [ p i − , p i ] is less than C , for i = 1 , . . . , n .Let x ′ = f ( x ), y ′ = f ( y ), and p ′ i = f ( p i ). Order A ′ so that x ′ ≤ y ′ . Let l ≥ p ′ l ≤ x ′ ≤ p ′ l +1 . Let m , l ≤ m ≤ n , be the smallest index so that p ′ m ≤ y ′ ≤ p ′ m +1 . So p ′ l +1 , . . . , p ′ m ∈ A ′ [ x ′ , y ′ ].Since d ( p ′ i , p ′ i +1 ) ≤ C , we have d ( p ′ l +1 , x ′ ) and d ( p ′ m , y ′ ) are both lessthan or equal to 2 Cλ . This lifts, by f , to give that d ( p l , x ) and d ( p m , y )are both less than or equal to 2 Cλ + C , and sodiam( A [ x, p l +1 ]) ≤ (2 Cλ + C ) λ and diam( A [ p m , y ]) ≤ (2 Cλ + C ) λ. Therefore, f ( A [ x, y ]) ⊂ f ( A [ x, p l +1 ] ∪ A [ p l +1 , p m ] ∪ A [ p m , y ]) ⊂ N ( { x ′ , y ′ } , (2 Cλ + C ) λ + C ) ∪ N ( A ′ [ x ′ , y ′ ] , Cλ ) ⊂ N ( A ′ [ f ( x ) , f ( y )] , (2 Cλ + C ) λ + C ) . (cid:3) The important point to note in Lemma 3.3 was the presence of the di-ameter constraint R allowing us to use a compactness type technique.Without this constraint we have various problems: our sequence ofcounterexamples still converges in some sense, but could give an un-bounded arc. Topological unzipping still works but the resulting arcswould not necessarily have a positive lower bound on separation.We can deal with the problem of no diameter bounds by dividing theproblem into two collections of non-interacting smaller problems. Tobe precise, given a λ -quasi-arc A , or even just a local λ -quasi-arc, we can use Lemma 3.3 on uniformly spaced out small subarcs of A (thatare genuine λ -quasi-arcs) with a sufficiently small ǫ value – this is thefirst collection of problems.Now the second collection of independent problems is how to jointogether two of these small splittings with two disjoint arcs havinguniform bound on their separation – but this a problem with boundeddiameter! So compactness arguments allow us to fix this and to removethe dependence of δ ⋆ on R in Lemma 3.3. Lemma 3.5.
Given < ǫ ≤ diam( X ) and an αǫ -local λ -quasi-arc A in X , where α ∈ (0 , is a constant, there exists δ ⋆ = δ ⋆ ( λ, L, N, α ) > such that for all δ < δ ⋆ we can split A into two arcs that ǫ -follow A and that are δǫ -separated.Proof. Without loss of generality we can rescale to ǫ = 1. As before,choose a linear order on A compatible with its topology. Let x be thefirst point in A , and y be the first point at distance D = α λ from x .(If there is no such point, diam( A ) ≤ = ǫ and so we can split A intotwo points that are ǫ -separated.) Label the next point at distance D from y by x . Continue in this manner with all jumps D until thelast label y n , with d ( x n , y n ) ∈ [ D , D ).Let D = D = α λ , and D = λ ( Lλ +2) D . We can control the in-teractions of the collection of sub-arcs of types A [ x i , y i ] and A [ y i , x i +1 ]:the D neighborhoods of two different such sub-arcs are disjoint out-side the collection of balls { B ( x i , D ) } ∪ { B ( y i , D ) } . This is becauseotherwise there are points z and z ′ in two different sub-arcs that satisfy d ( z, z ′ ) ≤ D < α , so the diameter of A [ z, z ′ ] is less than 2 λD < D – but A [ z, z ′ ] has to pass through the center of a D -ball that does notcontain z or z ′ , which is a contradiction.Now A [ x i , y i ] is a λ -quasi-arc, and we use Lemma 3.3 to create J i and J ′ i in a D neighborhood of A [ x i , y i ] that are δ -separated for some δ = δ ( λ, L, N, D ) >
0. By applying Theorem 2.1 to straighten thearcs, we may assume that they are λ ′ -quasi-arcs in a D neighborhoodof A [ x i , y i ] that are δ -separated, where λ ′ = λ ′ ( L, N, δ , D ).We want to join up the pair of arcs J i and J ′ i ending in B ( y i , D )to the arcs J i +1 and J ′ i +1 starting in B ( x i +1 , D ), without alteringthe setup outside the set Join( i ) = B ( y i , D ) ∪ N ( A [ y i , x i +1 ] , D ) ∪ B ( x i +1 , D ). Figure 3 shows this configuration. We will do this joiningin two stages: first, a topological joining that keeps the arcs disjoint,and second, a quantitative version that controls the separation of thearcs in the joining. Topological joining:
Join the endpoints of J i and J ′ i to the arc A inthe ball B ( y i , LD ) and the endpoints of J i +1 and J ′ i +1 to A in the ball ONFORMAL DIMENSION BOUNDS 13
Figure 3.
Joining unzipped arcs B ( x i +1 , LD ). Use the topological unzipping argument of Lemma 3.1to unzip A along this segment resulting in ‘wiring’ the pair ( J i , J ′ i ) tothe pair ( J i +1 , J ′ i +1 ) (not necessarily in that order) inside Join( i ). Thesearcs are disjoint, and so separated by some distance δ > Quantitative bound on δ : If there is no quantitative lower bound on δ then there are configurations (relabeling for convenience our joiningarcs) C n = ( X n , A n , J n , J ′ n , J n , J ′ n ) , where the best joining of the pair J n and J ′ n to the pair J n , J ′ n is atmost n -separated.But this configuration is precompact in the Gromov-Hausdorff topol-ogy as the X n are all N -doubling, and the arcs are all uniform quasi-arcs. (This is the importance of Tukia’s theorem.) So we can take asubsequence converging to a configuration C ∞ = ( X ∞ , A ∞ , J ∞ , J ′∞ , J ∞ , J ′∞ )in a suitable ball, and join the arcs using the topological method above,giving some valid rewiring with some positive separation δ ∞ >
0. Fol-lowing the proof of Lemma 3.3 we can lift this to C n for sufficientlylarge n retaining a separation of δ ∞ >
0, which is a contradiction forlarge n . Now since we have some δ ⋆ > i ), we can apply this pro-cedure for all i to create two arcs along A that are δ ⋆ -separated, for δ ⋆ depending only on λ , L , N , and α as desired. We assumed ǫ = 1,but rescaling to any ǫ gives the same conclusion with our resulting arcs δ ⋆ ǫ -separated. (cid:3) Bounding the conformal dimension from below
We now can use the unzipping results of Section 3 (Lemma 3.5) tocreate a Cantor set of arcs.By a Cantor set we mean the space Z = { , } N with an (ultra-)metric d σ (( a , a , . . . ) , ( b , b , . . . )) = exp( − (log(2) /σ ) inf { n | a n = b n } ) , where σ > Z, d σ ) has Hausdorffdimension σ , and is Ahlfors regular since there is a Borel probabilitymeasure ν σ on Z that satisfies r σ ≤ ν σ ( B ( z, r )) ≤ r σ , for all z ∈ Z and r ≤ diam( Z ).Returning to the metric space ( X, d ) of Theorem 1.1, we can nowprove Theorem 1.4.
Proof of Theorem 1.4.
Begin with any arc J ′ , assume it has endpointsone unit apart and apply Theorem 2.1 to J ′ and ǫ = to get J ∅ , a λ -quasi-arc on scales below α . Let our scaling factor be β = αδ ⋆ λ ≤ .We can assume that, for a given n , we have a collection of λ -quasi-arcs on scales below β n , written as { J a a ...a n | a i ∈ { , } , ≤ i ≤ n } ,and that these arcs are β n separated.For each J a a ...a n , we split it into two arcs using Lemma 3.5 appliedto ǫ = β n , then straighten each arc using Theorem 2.1 with ǫ = δ ⋆ β n to get two new arcs J a a ...a n and J a a ...a n that are λ -quasi-arcson scales below αδ ⋆ β n ≥ β n +1 , and are δ ⋆ β n ≥ β n +1 separated. Infact, all the arcs created at this stage are β n +1 separated as the newarcs arising from different arcs in the previous generation can only get2 (cid:0) β n + δ ⋆ β n (cid:1) < β n closer, thus remaining at least β n +1 apart.At this point it is useful to record the following. Lemma 4.1. If J is a λ -quasi-arc on scales below ǫ , and we have anarc J ′ ⊂ N ( J, ǫ ) , whose endpoints are ǫ close to those of J , then wemust have J ⊂ N ( J ′ , λǫ ) . In particular, d H ( J, J ′ ) ≤ λǫ . Given a sequence a = ( a , a , . . . ) ∈ { , } N , the sequence of arcs J ∅ ,J a , J a a , . . . is Cauchy in the Hausdorff metric (using Lemma 4.1),and hence convergent to J a a ... = J a , a set of diameter at least . A ONFORMAL DIMENSION BOUNDS 15 priori, this set need not be an arc, but only compact and connected.(This is actually enough to apply the argument of Pansu’s lemma.)However, for each n we know that J a a ...a n is a β n -local λ -quasi-arc that β n -follows J a a ...a n − , and we know that β < min (cid:8) λ , (cid:9) . Usingthese facts, [Mac08, Lemma 2.2] shows that J a is a λ ′ -quasi-arc, with λ ′ = λ ′ ( β, L, N ) = λ ′ ( L, N ), that β n -follows J a a ...a n for each n .(Finding quasi-arcs in the limit is not unexpected since on each scalethe limit set will look like the quasi-arc approximation on the samescale.)If we set M ( X ) to be the set of all closed sets in X , we can definea map F : Z → M ( X ) by F ( a ) = J a . Let J = F ( Z ) be the image ofthis map and choose the metric d σ for Z , σ = − log(2)log( β ) >
0. It remainsto show that F : ( Z, d σ ) → ( M ( X ) , d H ) is a bi-Lipschitz embedding.Take a = ( a , a , . . . ) , b = ( b , b , . . . ) ∈ Z . Then d σ ( a, b ) ∈ ( β n +1 , β n ]if and only if a i = b i for 1 ≤ i < n and a n = b n . By construction, anda geometric series, J a ⊂ N ( J a ...a n , β n ), and so as n stage arcs are β n separated, we have(1) d H ( J a , J b ) ≥ d ( J a , J b ) ≥ β n ≥ d σ ( a, b ) . Conversely, applying the triangle inequality and Lemma 4.1, we have(2) d H ( J a , J b ) ≤ d H ( J a , J a ...a n − )+ d H ( J b ...b n − , J b ) ≤ λβ n − ≤ λβ d σ ( a, b ) , so F is bi-Lipschitz, quantitatively.As a final remark, note that there is a natural measure µ σ = F ∗ ( ν σ )on J . The estimates (1) and (2) imply that, for any ball B ( x, r ) ⊂ X ,the set { J a ∈ J | J a ∩ B ( x, r ) = ∅} is measurable (in fact open), andif two arcs J a and J b both meet this ball, we have 2 r ≥ d ( J a , J b ) ≥ d σ ( a, b ), and so µ σ { J a ∈ J | J a ∩ B ( x, r ) = ∅} ≤ σ r σ . (cid:3) We now prove our main theorem.
Proof of Theorem 1.1.
The construction of Theorem 1.4 gives a lowerbound for conformal dimension by virtue of the following lemma ofPansu [Pan89b, Lemma 6.3]. This version is due to Bourdon.
Lemma 4.2 ([Bou95, Lemma 1.6]) . Suppose that ( X, d ) is a uniformlyperfect, compact metric space containing a collection of arcs C = { γ i | i ∈ I } whose diameters are bounded away from zero. Suppose further thatwe have a Borel probability measure µ on C and constants A > , σ ≥ such that, for all balls B ( x, r ) in X , the set { γ ∈ C| γ ∩ B ( x, r ) = ∅} is µ -measurable with measure at most Ar σ . Then the conformal dimensionof X is at least στ − σ , where τ is the packing dimension of X , andin fact τ − σ ≥ . In our case X may be non-compact, but it is proper and all arcs γ ∈ C lie in some fixed (compact) ball in X . The packing dimensionof X is finite and bounded from above by a constant derived from thedoubling constant N . Furthermore, X is connected, so it is certainlyuniformly perfect.Following Theorem 1.4, we apply Lemma 4.2 with C = J , µ = µ σ and A = 4 σ , where σ depends only on L and N , to find a lower boundfor the conformal dimension of C = C ( L, N ) > (cid:3) We now apply our theorem to the case of conformal boundaries ofhyperbolic groups.
Proof of Corollary 1.2.
In [BK05b, Proposition 4], Bonk and Kleinershow that ∂ ∞ G with some visual metric d is compact, doubling andlinearly connected. It remains to show that ( X, d ) = ( ∂ ∞ G, d ) is annu-larly linearly connected, but this follows by a proof similar to that ofBonk and Kleiner’s proposition.Suppose (
X, d ) is not annularly linearly connected. Then there isa sequence of annuli A n = A ( z n , r n , r n ) containing points x n and y n such that there is no arc joining x n to y n inside A ( z n , n r n , nr n ). As X is compact we have r n →
0; otherwise, there would be a subsequence n j → ∞ as j → ∞ with r n j > ǫ > ǫ . In this case, takefurther subsequences so that r n j → r ∞ ∈ [ ǫ, diam( X )], z n j → z ∞ , x n j → x ∞ , and y n j → y ∞ . Then a contradiction follows from the factthat z ∞ is not a local cut point, so we must have r n → X, r n d, z n ). By dou-bling, this subconverges to a limit ( W, d W , z ∞ ) with respect to pointedGromov-Hausdorff convergence. By [BK02, Lemma 5.2], W is homeo-morphic to ∂ ∞ G \ { p } for some p , and so z ∞ cannot be a local cut pointin W . So we can connect the components of A ( z ∞ , . , .
1) in W \ z ∞ byfinitely many compact sets, and these must lie in some A ( z ∞ , /M, M )for 1 ≤ M < ∞ . For sufficiently large n we can lift these connectingsets to A ( z n , M r n , M r n ), contradicting our hypothesis.In conclusion, ∂ ∞ G is annularly linearly connected, doubling andcomplete, and so Theorem 1.1 gives that the conformal dimension of ∂ ∞ G is strictly greater than one. (cid:3) ONFORMAL DIMENSION BOUNDS 17
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