In this note, we use some combinatorial modulus on the boundary of a
right-angled hyperbolic building to control its conformal dimension. The lower
bound obtained is optimal in the case of Fuchsian buildings

Clais, Antoine

English

arXiv

Antoine Clais

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 355 (2017) 819–823Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comGeometry/Group theoryConformal dimension on boundary of right-angled hyperbolic buildingsDimension conforme du bord d’un immeuble hyperbolique à angles droitsAntoine ClaisTechnion, Department of Mathematics, 32000 Haifa, Israela r t i c l e i n f o a b s t r a c tArticle history:Received 6 May 2016Accepted after revision 9 June 2017Available online 10 July 2017Presented by the Editorial BoardIn this note, we use some combinatorial modulus on the boundary of a right-angled hyperbolic building to control its conformal dimension. The lower bound obtained is optimal in the case of Fuchsian buildings.© 2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éDans cette note, on utilise des modules combinatoires sur le bord d’un immeuble hyperbolique à angles droits pour encadrer sa dimension conforme. La borne inférieure obtenue est optimale dans le cas des immeubles fuchsiens.© 2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. IntroductionThe conformal dimension is a quasi-isometric invariant of Gromov hyperbolic spaces that has been introduced by P. Pansu in [12]. Since then it has become a major tool used to study quasi-conformal properties of boundaries of hyperbolic groups in relation with rigidity phenomena. In particular, in [1], it plays a key role in proving that right-angled Fuchsian buildings satisfy a Mostow type rigidity theorem. We refer to [9] and [8] for surveys concerning the connection between the conformal dimension and Mostow type rigidity results and to [10] for a survey concerning the conformal dimension in the more general context of self-similar spaces.In this note, we give bounds to the conformal dimension of the boundary of a right-angled hyperbolic building. This is obtained by simple computations not on the boundary of the building, but on the boundary of an apartment.The motivation of this work is that a good understanding of the conformal dimension of these buildings could lead to new rigidity results. Some of them are indeed highly suspected to be rigid.E-mail address: aclais@tx.technion.ac.il.http://dx.doi.org/10.1016/j.crma.2017.06.0061631-073X/© 2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.820 A. Clais / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 819–8232. RecallsLet (Z , d) and (Z ′, d′) be two compact arcwise connected metric spaces. The cross-ratio of four distinct points a, b, c, d ∈ Zis[a : b : c : d]Z = d(a,b)d(a, c)· d(c,d)d(b,d).An homeomorphism f : Z −→ Z ′ is quasi-Moebius (QM) if there exists an homeomorphism φ :R+ −→ R+ such that for any quadruple of distinct points a, b, c, d ∈ Z[ f (a) : f (b) : f (c) : f (d)]Z ′ ≤ φ([a : b : c : d]Z ).In this case, f −1 is also QM and Z and Z ′ are said quasi-Moebius equivalent (QM equivalent). Now we assume that Z is a Q -Ahlfors-regular (AR) for Q > 1. This means that there exists a constant C > 1 such that for any 0 < R ≤ diam Z and any R-ball B ⊂ Z one hasC−1 · R Q ≤ Hd(B) ≤ C · R Q ,where Hd(·) denotes the Hausdorff measure of (Z , d). Notice that, under this assumption, Q is equal to the Hausdorff dimension dimH(Z , d) of (Z , d). The Ahlfors-regular conformal gauge of (Z , d) is defined as follows:Jc(Z ,d) := {(Z , δ) : (Z , δ) is AR and is QM equivalent to (Z ,d)}.Definition 2.1. The Ahlfors-regular conformal dimension of ∂ isConfdim(Z ,d) := inf{dimH(Z , δ) : (Z , δ) ∈ Jc(Z ,d)}.In the rest of the note we will simply call it the conformal dimension. As the topological and the Hausdorff dimen-sions are respectively invariant under homeomorphisms and bi-Lipschitz maps, the conformal dimension is invariant under quasi-Moebius maps. The inclusions between these three classes of maps imply the following inequalities:dimT (Z) ≤ Confdim(Z ,d) ≤ dimH(Z ,d),where dimT (Z) stands for the topological dimension of Z .We recall that all the visual metrics on the boundary of a hyperbolic space are quasi-Moebius homeomorphic to each other and AR. In particular, the conformal dimension is a quasi-isometric invariant of a hyperbolic space. This will be the context of this note.Combining ideas of G.D. Mostow, P. Pansu, and M. Bourdon with a theorem of M. Bonk and B. Kleiner (see [1, Theo-rem 1.3]), it is known that if the conformal dimension of the boundary of a CAT(-1) group is achieved in the conformal gauge, then the underlying CAT(-1) space satisfies a Mostow-type rigidity theorem (see [8, Théorème 5.11]).Hence, the conformal dimension is a very powerful tool, but, unfortunately, it is also very hard to compute.3. The resultThe goal of this note is to prove the theorem below that relates the conformal dimension of the boundary of a building to the conformal dimension of the boundary of an apartment. We refer to [6] for generalities concerning Coxeter groups and buildings.Let G denote a finite simplicial graph i.e G(0) is finite, each edge has two different vertices, and G contains no double edge. We denote by G(0) = {v1, . . . , vn} the vertices of G and we set S = {s1, . . . , sn}. If for i = j the corresponding vertices vi, v jare connected by an edge, then we write vi ∼ v j . We denote by W the right-angled Coxeter group whose relation graph is G , namelyW =〈si ∈ S|s2i = 1, si s j = s j si if vi ∼ v j〉.For q ≥ 2 we denote by q the group defined by the following presentationq =〈si ∈ S|sqi = 1, si s j = s j si if vi ∼ v j〉.In the rest of this note, we will assume that q is infinite hyperbolic with arcwise connected boundary ∂q . This assumption is in fact only an assumption on the graph G (see [11] and [7]). We also equip ∂q with a visual metric d.We recall that q acts by isometry properly discontinuously and cocompactly on q the building of type (W , S) and of constant thickness q. As a consequence, the boundaries ∂q and ∂q are canonically identified by a q-equivarient quasi-Moebius homeomorphism. We also recall that q is a CAT(-1) metric space.In the following, for g ∈ q , we designate by |g| the distance, for the word metric on q relative to the generating set Sq , between e and g . Then τ (q) = lim supk 1k log(#{g ∈ q : |g| ≤ k}) is the growth rate of q . In the rest of the note, we will write Q (q) = Confdim(∂q). Notice that, in particular, Q (2) = Confdim(∂W ).A. Clais / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 819–823 821Theorem 3.1. There exists a constant C > 0 independent of q such thatQ (2) · (1 + log(q − 1)τ (2)) ≤ Q (q) ≤ C log(q − 1).4. Consequences of Theorem 3.1In general, the conformal dimension Q (2) of ∂W is unknown. However, the topological dimension of ∂W is easy to read in the graph G (see [7]). Hence, one can use the inequality Confdim(∂W ) ≥ dimT (∂W ), to obtain an explicit lower bound.The only hyperbolic buildings for which we can compute the conformal dimension of their boundaries are Fuchsian buildings. In this case, the lower bound of Theorem 3.1 is optimal (see [2, Théorème 1.1]). We can wonder if this bound is optimal for other examples. The following example is particularly interesting because it should lead to a new rigidity result.Example 1. Let D be the regular right-angled dodecahedron in H3 and W D be the right-angled Coxeter group generated by the reflections about the faces of D . Then ∂W D is quasi-Moebius homeomorphic to the Euclidean sphere S2 and Q (2) = 2.For q ≥ 3, the corresponding building q has all chances to be Mostow rigid. Indeed, Fuchsian buildings (the analogues of q in dimension 2) are rigid, whereas their apartments are not. These are hyperbolic planes H2. On the other hand, the apartments of q are rigid, they are hyperbolic 3-spaces H3. Hence, the rigidity of the apartments should be increased by the building structure. Moreover, in [5] it is proved that ∂q satisfies the combinatorial Loewner property. This property is conjecturally equivalent to admitting a metric realizing the conformal dimension (see [9, Conjecture 7.5]). A next step towards the rigidity of this building would be to know if the lower bound Q (q) ≥ 2 + 2 log(q−1)τ (2) is sharp.This example has an analogue in dimension 4 if we replace the dodecahedron in H3 by the right-angled regular 120-cell in H4. The boundary of the building obtained also satisfies the combinatorial Loewner property and for the same reasons has also all the chances to be rigid.Finally, an immediate computation shows that the growth rates τ (q) and τ (2) are related by the formula τ (q) = τ (2) +log(q −1). On the other hand, growth rates of Coxeter groups can be computed by hand. For instance, if W D is the reflection group of Example 1, then τ (2) = log(4 + √15).5. Combinatorial modulusFor a complete introduction on combinatorial modulus on boundaries of hyperbolic groups we refer to [3]. In this note, we will restrict to the example given by the group ∂q equipped with a visual metric d.For k ≥ 0 and κ > 1, a κ-approximation of (∂q, d) on scale k is a finite covering Gk by open subsets such that for any v ∈ Gk there exists zv ∈ v satisfying the following properties:– B(zv , κ−12−k) ⊂ v ⊂ B(zv , κ2−k),– ∀v, w ∈ Gk with v = w one has B(zv , κ−12−k) ∩ B(zw , κ−12−k) = ∅.A sequence {Gk}k≥0 is called a κ-approximation of (∂q, d).Now we fix the approximation {Gk}k≥0. Throughout this note, we will call a curve in ∂q a continuous map γ : [0, 1] −→∂q and we will denote by γ its image. Let ρ : Gk −→ [0, +∞) be a positive function and γ be a curve in ∂q . The ρ-lengthof γ isLρ(γ ) =∑γ ∩v =∅v∈Gkρ(v).For p ≥ 1, the p-mass of ρ is Mp(ρ) = ∑v∈Gkρ(v)p . Let F be a non-empty set of curves in ∂q . We say that the function ρis F -admissible if Lρ(γ ) ≥ 1 for any curve γ ∈F . For p ≥ 1, the Gk-combinatorial p-modulus of F isModp(F, Gk) = inf{Mp(ρ)},where the infimum is taken over the set of F -admissible functions and with Modp(∅, Gk) := 0.6. Two ingredients for the proofThe first ingredient of the proof is that the conformal dimension is a critical exponent for the combinatorial modulus. Let d0 > 0 be a small constant compared with the geometric constants of (∂q, d) and let F0 be the set of all the curves in ∂q of diameter larger than d0. Under this assumptionConfdim(∂q) = inf{p ∈ [1,+∞) : lim Modp(F0, Gk) = 0}. (1)k→+∞822 A. Clais / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 819–823This is the immediate application of an unpublished theorem due to S. Keith and B. Kleiner (see [4, Theorem 1.2.] for a proof in a more general context).The second ingredient of the proof is a control of the combinatorial modulus on the boundary of the building by the combinatorial modulus on the boundary of an apartment established in [5], which we recall here briefly. Let G Wk and Gkbe two κ-approximations of respectively ∂W and ∂ constructed as in [5, Subsection 8.2]. We recall that by construction, there exists λ ≥ 1 such that, for any k ≥ 0#G Wk ≤ #{g ∈ W : |g| ≤ k} ≤ λ · #G Wk .In particular, this implies that τ (2) = lim supk→∞ 1k log(#G Wk ). Now we designate by modp(·, GWk ) and Modp(·, Gk) the com-binatorial modulus computed respectively in ∂W and in ∂q . Let d0 > 0 be a small constant compared with the geometric parameters of ∂W and ∂q and let FW0 and F0 be the set of all the curves respectively in ∂W and ∂q of diameter larger than d0. Under these assumptions, according to [5, Theorem 9.1], for any p ≥ 1, there exists a constant D ≥ 1 such that, for every k ≥ 1,D−1 · Modp(F0, Gk) ≤ (q − 1)k · modp(FW0 , G Wk ) ≤ D · Modp(F0, Gk). (2)As an immediate consequence of the relations (1) and (2),Q (q) = inf{p ∈ [1,+∞) : limk→+∞(q − 1)k · modp(FW0 , G Wk ) = 0}. (3)7. Right-hand-side inequalityWe use the equality (3) in proving Theorem 3.1. Throughout the proof, {ρk}k≥0 is a sequence of FW0 -admissible functions.Comparing the diameters of the elements of Gk with d0, one obtains that there exist two constants K > 0 and 0 < λ < 1independent of k and q such that, up to changing the sequence {ρk}k≥0, for all k ≥ 1 and all w ∈ G Wkρk(w) ≤ K · λk.Hence(q − 1)k∑w∈G Wkρk(w)p ≤ K p · #G Wk · [λp(q − 1)]k.As we recalled, τ (2) = lim supk→∞ 1k log(#G Wk ), thus Q (q) ≤ 1log 1/λ (τ (2) + log(q − 1)).8. Left-hand-side inequalityNow we set G = ⋃k≥0GWk and for w ∈ G we write |w| = k such that w ∈ G Wk . For a sequence {ρk}k≥0 of FW0 -admissible function, let ρ : w ∈ G −→ ρ|w|(w) ∈ R+ . Now we observe thatif∑w∈G(q − 1)|w|ρ(w)p < ∞ then p ≥ Q (q).We define the functionPρ(s) := inf{p > 0|∑w∈G(q − 1)s|w|ρ(w)p < ∞}and we study the function Pρ to compute the lower bound for Q (q).To start, one has Pρ(0) ≥ Q (2) and, up to changing {ρk}k≥0, we can choose Pρ(0) arbitrarily close to Q (2). Then, identically, one has Pρ(1) ≥ Q (q) and, up to changing {ρk}k≥0, we can choose Pρ(1) arbitrarily close to Q (q).Now we set s0 = sup{s ∈R| ∑w∈G(q − 1)s|w| < ∞}. Clearly, s0 < 0 and is such that Pρ(s0) = 0. Indeed, as we said in the preceding paragraph, there exist two constants K > 0 and 0 < λ < 1 independent of k and q such that, up to changing the sequence {ρk}k≥0, for all k ≥ 1 and all w ∈ G Wkρ(w) ≤ K · λ|w|.Hence, if for all ε > 0∑(q − 1)−ε|w|.(q − 1)s0|w| < ∞.w∈GA. Clais / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 819–823 823Then for all ε′ > 0 small enough, one has∑w∈Gρ(w)ε′(q − 1)s0|w| ≤ K ε′ ·∑w∈G(q − 1)−ε|w|.(q − 1)s0|w| < ∞.On the other hand, s0 = sup{s ∈R| ∑k∈N #G Wk (q − 1)sk < ∞}. As we recalled, τ (2) = lim supk 1k log(#G Wk ) thuss0 = − τ (2)log(q − 1) .Now we check that the function Pρ is convex on [s0, +∞). In other word, we check that for all t ∈ [0, 1], for all [a, b] ⊂ [s0, +∞) and for all ε > 0 one has∑w∈G(q − 1)(ta+(1−t)b)|w|ρ(w)(t Pρ(a)+(1−t)Pρ(b))+ε < ∞.Indeed, for all ε > 0∑w∈Gxw :=∑w∈G(q − 1)a|w|ρ(w)Pρ(a)+ε < ∞ and∑w∈Gyw :=∑w∈G(q − 1)b|w|ρ(w)Pρ(b)+ε < ∞.Hence {x1/pw } ∈ p and {y1/qw } ∈ q with p = 1t and q = 11−t and by Hölder’s inequality:(∑w∈Gxw)1/p .(∑w∈Gyw)1/q ≥∑w∈G((q − 1)a|w|ρ(w)Pρ(a)+ε)t .((q − 1)b|w|ρ(w)Pρ (b)+ε)1−t,≥∑w∈G(q − 1)(ta+(1−t)b)|w|ρ(w)(t Pρ(a)+(1−t)Pρ(b))+ε .Finally, by convexity, for all t < 0 one has Pρ(ts0) ≥ t Pρ(s0) + (1 − t)Pρ(0). In particular, for t = 1/s0 one hasQ (q) ≥ Q (2) · (1 − 1s0) = Q (2) · (1 + log(q − 1)τ (2)).AcknowledgementsThe author thanks Marc Bourdon for his precious suggestions and advices.References[1] M. Bonk, B. Kleiner, Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary, Geom. Topol. 9 (2005) 219–246.[2] M. Bourdon, Immeubles hyperboliques, dimension conforme et rigidité de Mostow, Geom. Funct. Anal. 7 (1997) 245–268.[3] M. Bourdon, B. Kleiner, Combinatorial modulus, the combinatorial Loewner property, and Coxeter groups, Groups Geom. Dyn. 7 (2013) 39–107.[4] M. Carrasco Piaggio, On the conformal gauge of a compact metric space, Ann. Sci. Éc. Norm. Supér. (4) 46 (2013) 495–548.[5] A. Clais, Combinatorial modulus on boundary of Right–Angled hyperbolic buildings, Anal. Geom. Metr. Spaces 4 (1) (2016).[6] M. Davis, The Geometry and Topology of Coxeter Groups, Princeton University Press, Princeton, NJ, USA, 2008.[7] M. Davis, J. Meier, The topology at infinity of Coxeter groups and buildings, Comment. Math. Helv. 77 (2002) 746–766.[8] P. Haïssinsky, Géométrie quasiconforme, analyse au bord des espaces métriques hyperboliques et rigidités [d’après Mostow, Pansu, Bourdon, Pajot, Bonk, Kleiner. . . ], in: Séminaire Bourbaki, Astérisque 2007/2008 (2009).[9] B. Kleiner, The asymptotic geometry of negatively curved spaces: uniformization, geometrization and rigidity, in: International Congress of Mathemati-cians. Vol. II, 2006, pp. 743–768.[10] J. Mackay, J. Tyson, Conformal Dimension, American Mathematical Society, Providence, RI, USA, 2010.[11] J. Meier, When is the graph product of hyperbolic groups hyperbolic?, Geom. Dedic. 61 (1996) 29–41.[12] P. Pansu, Dimension conforme et sphère à l’infini des variétés à courbure négative, Ann. Acad. Sci. Fenn., Ser. A I Math. 14 (1989) 177–212.

Comptes Rendus Mathematique

Conformal dimension on boundary of right-angled hyperbolic buildings

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Conformal dimension on boundary of right-angled hyperbolic buildings

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