Homological dimension and critical exponent of Kleinian groups 7 0 0 2 Michael Kapovich n a J February 2, 2008 8 2 ] R Abstract G We prove an inequality between the relative homological dimension of a . h Kleinian group Γ Isom(Hn) and its critical exponent. As an application at of this result we sh⊂ow that for a geometrically finite Kleinian group Γ, if the m topological dimension of the limit set of Γ equals its Hausdorff dimension, then [ the limit set is a round sphere. 1 v 7 1 Introduction 9 7 1 0 One of the frequent themes in the theory of Kleinian groups is establishing a relation 7 between the abstract algebraic properties of a Kleinian group and its geometric prop- 0 / erties, determined by its action on the hyperbolic space. Ahlfors finiteness theorem h t and Mostow rigidity theorem are among the most important examples of such rela- a m tion. In this paper we establish a relation between two invariants of a Kleinian group: : Virtual homological dimension (an algebraic invariant) and the critical exponent (a v geometric invariant). We refer the reader to Section 2 for the precise definitions. i X Given a Kleinian group Γ Isom(Hn), consider the set of its maximal virtually r ⊂ P a abelian subgroups of virtual rank 2, i.e. the elements of are maximal subgroups ≥ P which contain a subgroup isomorphic to Z2. Form the maximal subset Π := Π ,i I i { ∈ } ⊂ P of pairwise nonconjugate elements of . In other words, Π consists of representatives P of cusps of rank 2 in Γ. ≥ We let vhd (Γ,Π) and vcd (Γ,Π) denote the virtual homological and cohomologi- R R cal dimension of Γ relative to Π, where R is a commutative ring with a unit. (Instead of working with virtual dimensions, one can use the (co)homological dimension with respect to fields of zero characteristic, or, more generally, rings where the order of every finite subgroup of Γ is invertible.) Let δ(Γ) be the critical exponent of Γ. Our main result is 1 Theorem 1.1. Suppose that Γ is a virtually torsion-free Kleinian group. Then vhd (Γ,Π) 1 δ(Γ). R − ≤ Corollary 1.2. Suppose that the pair (Γ,Π) has finite type, e.g. Γ admits a finite K(Γ,1) and the set Π is finite. Then cd (Γ,Π) 1 δ(Γ). R − ≤ One, therefore, can regard these results as either nontrivial lower bounds on the critical exponent, or as vanishing theorems for relative (co)homology groups of Γ with arbitrary twisted coefficients. These results also can be viewed as generalizing the classical inequality dim(Z) dim (Z) H ≤ for compact metric spaces Z, see [19]. Here dim(Z) is the topological dimension and dim (Z) is the Hausdorff dimension. H As an application of Corollary 1.2 we prove Theorem 1.3. Suppose that Γ Isom(Hn) is a nonelementary geometrically finite ⊂ group so that the Hausdorff dimension of its limit set equals its topological dimension d. Then the limit set of Γ is a round d-sphere, i.e. Γ preserves a d+1-dimensional subspace H Hn and H/Γ has finite volume. ⊂ This theorem was first proved by Rufus Bowen [12] for convex-cocompact quasi- fuchsian subgroups of Isom(H3). Bowen’s theorem was extended by Bishop and Jones [9] to subgroups of Isom(H3) with parabolic elements. Bowen’s result was generalized by Chenbo Yue [29] to convex-cocompact subgroups of Isom(Hn) whose limit sets are topological spheres, although his argument did not need the latter assumption. Note that the arguments of Yue do not work in the presence of parabolic elements. For cocompact discrete groups of isometries of CAT( 1) spaces, an analogue of Theorem − 1.3 was proved by Bonk and Kleiner [10], see also the work of Besson, Gallot and Courtois [3]. The latter paper was the inspiration for our work. Conjecture 1.4. Suppose that Γ is a finitely-generated Kleinian group in Isom(Hn). Then: 1. d = vcd (Γ,Π) 1 δ(Γ). R − ≤ 2. In the case of equality, Γ is geometrically finite and its the limit set is a round d-sphere in Sn−1. Another application of our main theorem is the following property of groups with small critical exponent: Corollary 1.5. Suppose that δ(Γ) < 1 and Γ is of type FP , e.g., is finitely-presented. 2 Then Γ is virtually free. Problem 1.6. (Cf. Theorem 1.3 in [9].) Is it true that every finitely-generated Kleinian group Γ with δ(Γ) < 1 is geometrically finite? Is it true that such group is a classical Schottky-type group? 2 The proofs of our results are generalizations of the proofs due to Besson, Courtois and Gallot in [3]. Our main contribution in comparison to their paper is treatment of arbitrary coefficient modules, working with relative homology groups and handling manifolds whose injectivity radius is not bounded from below. The most nontrivial technical ingredient of our paper is existence of the natural maps introduced in [3] and their properties established in that paper. In the case of finitely-generated Kleinian subgroups Γ Isom(H3), our main ⊂ theorem easily follows from the well-known facts about Γ. It suffices to consider the case when Γ is torsion-free. If δ(Γ) = 2, then Theorem 1.1 states that vhd(Γ,Π) 3. ≤ The letter inequality immediately follows from the fact that the hyperbolic mani- fold H3/Γ is a 3-dimensional Eilenberg-MacLane space for Γ. Assume therefore that δ(Γ) < 2. Then it follows from the solution of the Tameness Conjecture [1], [15] (which, in turn, implies Ahlfors’ measure zero conjecture) and [9], that Γ is geomet- rically finite. Therefore either Γ is a Schottky-type group or it contains a finitely- generated quasi-fuchsian subgroup Φ Γ, whose limit set is a topological circle. In ⊂ the later case, 2 vhd (Γ,Π) vhd (Φ,Π Φ) = 2, R R ≥ ≥ ∩ while δ(Γ) δ(Φ) 1. ≥ ≥ This implies the inequality 1 = 2 1 = vhd (Γ,Π) 1 1 δ(Γ). R − − ≤ ≤ If Γ is a Schottky-type group, then Γ = F Π ... Π , ∼ k 1 m ∗ ∗ ∗ where Π Π for i = 1,...,m. Therefore vhd (Γ,Π) = 1 and Theorem 1.1 trivially i R ∈ follows. Sketch of the proof of Theorem 1.1. Let ǫ be a positive number which is smaller thantheMargulisconstantµ forHn. Letδ := δ(Γ). WeassumethatΓistorsion-free. n We sketch the proof under the following assumption: There exists a thick triangulation of the hyperbolic manifold M = Hn/Γ, i.e. a triangulation T and a number L < , so that every i-simplex in T not contained in ∞ the ǫ-thin part M of M is L-bilipschitz diffeomorphic to the standard Euclidean (0,ǫ] i-simplex. (Existence of such triangulation was recently proved by Bill Breslin [13] for n = 3.) Suppose that hd (Γ,Π) > δ + 1. Then for some q > δ + 1, there exists a flat R bundle V over the manifold M, so that H (M,M ;V) = 0. q (0,ǫ] 6 3 Pick a chain ζ C (M;V) which projects to a nonzero class [ζ] in H (M,M ;V). q q (0,ǫ] ∈ ˆ We then extend ζ to the ǫ-thin part of M, to a locally finite absolute cycle ζ of finite volume. Besson, Courtois and Gallot in [3] proved existence of a natural map F : M M which is (properly) homotopic to the identity and satisfies → q δ +1 ˆ ˆ vol(F (ζ)) vol(ζ). # ≤ (cid:18) q (cid:19) Since q > δ +1, the locally finite cycle ζˆ := Fk(ζˆ) satisfies k # ˆ lim vol(ζ ) = 0. k k→∞ Then we use the deformation lemma of Federer and Fleming to deform (for large k) ˆ ˆ the cycle ζ to a locally finite cycle ξ which is supported in the q 1-skeleton of T k k away from M . Therefore ξˆ determines zero homology class in−H (M,M ;V). (0,ǫ] k q (0,ǫ] Since Fk is properly homotopic to the identity (with uniform control on the length of the tracks of the homotopy) we conclude that [ζ] is trivial as well, which is a contradiction. Since the existence of a thick triangulationis not proven in general, we use instead a map η from M to a simplicial complex X, which is the nerve of anappropriate cover ofM. Themapη isL -Lipschitzontheκ-thickpartofM forevery κ > 0. Thisallows κ us to do the deformation arguments in X rather than in T. This line of arguments is borrowed from [18, 5.32]. § Acknowledgements. This work was partially supported by the NSF grant DMS 0405180. Most of this paper was written when the author was visiting the Max Plank InstituteforMathematicsinBonn. IamgratefultoG´erardBessonandGillesCourtois for sharing with me an early version of [3] and to Leonid Potyagailo for motivating discussions. 2 Preliminaries 2.1 Geometric preliminaries Basics of Kleinian groups. We let Hn denote the hyperbolic n-space, Sn−1 the ideal boundary of Hn, and Isom(Hn) the isometry group of Hn. A Kleinian group is a discrete isometry group of Hn. The limit set of a Kleinian group Γ is denoted Λ(Γ). A Kleinian group Γ is called elementary if its limit set contains at most 2 points. A Kleinian group is elementary if and only if it is virtually abelian. We let Hull(Λ(Γ)) Hn ⊂ denote the convex hull of Λ(Γ) in Hn. Let Γ Isom(Hn) be a Kleinian group, x Hn be a point and ǫ be a positive real ⊂ ∈ number. Let Γ Γ x,ǫ ⊂ 4 denote the subgroup generated by the elements γ Γ such that ∈ d(x,γ(x)) ǫ. ≤ Then, according to Kazhdan–Margulis lemma, for every n there is a constant µ > 0, called the Margulis constant, such that Γ is elementary, for every Kleinian n x,µn subgroup Γ Isom(Hn) and every point x Hn. ⊂ ∈ Thick-thin decomposition of hyperbolic manifolds. For a point x in a Riemannian manifold M (possibly with convex boundary) define InRad (x) M to be the injectivity radius of M at x. Then the function InRad is 1-Lipschitz, i.e., M it satisfies InRad (x) InRad (x′) d(x,x′). (1) M M | − | ≤ Suppose that M is a metrically complete connected hyperbolic manifold with convex boundary. Let M˜ denote the universal cover of M. For 0 < ǫ < µ consider n the thick-thin decomposition M = M M . (0,ǫ] [ǫ,∞) ∪ Here thin part K = M of M is the closure of the set of points x M, such that (0,ǫ] ∈ there exists a homotopically nontrivial loop γ based at x, whose length is < ǫ. x Let K , i J N, denote the connected components of K. i ∈ ⊂ Lemma 2.1. Each K is covered by a contractible submanifold K˜ in Hn. i i Proof. We identify π (K ) with an elementary subgroup Π Γ. Then K˜ = K˜ (ǫ) is 1 i i i i ⊂ the union K˜ (ǫ) = K˜ (γ), i ǫ [ γ∈Πi\{1} where K˜ (γ) = z M˜ : d(z,γ(z)) ǫ . ǫ { ∈ ≤ } ˜ Each K (γ) is convex, since the displacement function of γ is convex. Of course, the ǫ union of convex sets need not be convex and K is, in general, not convex. We first i consider the case when Π is a cyclic hyperbolic subgroup. Let A = A denote the i i ˜ common axis of the nontrivial elements of Π . Then A is contained in each K (γ). It i ǫ follows that K˜ := K˜ (ǫ) is star-like with respect to every point of A. Therefore K˜ is i i i contractible. If Π is parabolic, this argument of course does not apply. Let ξ = ξ denote i i ˜ the fixed point of Π . Then K is star-like with respect to ξ. Therefore, every map i i f : Sk K˜ (ǫ) can be homotoped to a map f : Sk K˜ (κ) along the geodesics i κ i → → asymptotic to ξ, where κ and d(κ) := diam(f (Sk)) κ 5 can be chosen arbitrarily small. Then f (Sk) bounds a ball f (Bk+1) within d(κ) κ κ from the image of f . Thus κ f (Bk+1) K˜ (κ+2d(κ)). κ i ⊂ ˜ By choosing κ so that κ+2d(κ) < ǫ, we conclude that π (K ) = 0 for all k. k i Therefore each K = K(Π ,1) is an Eilenberg-MacLane space for its fundamental i i group Π . i Critical exponent of a Kleinian group. Let Γ Isom(Hn) be a Kleinian ⊂ group. Consider the Poincar´e series f = e−sd(γ(o),o), s X γ∈Γ where o Hn is a base-point and d is the hyperbolic metric on Hn. Then the critical ∈ exponent of Γ is δ(Γ) = inf s : f < . s { ∞} Critical exponent has several alternative descriptions. Define N(R) := # x Γ o : d(x,o) R . { ∈ · ≤ } Then δ(Γ) is the rate of exponential growth of N(R), i.e. log(N(R)) δ(Γ) = limsup , R R→∞ see [23]. Lastly, the critical exponent can be interpreted in terms of the geometry of the limit set of Γ. Theorem 2.2. (See [9, 23, 25, 28].) For every Kleinian group Γ Isom(Hn), we ⊂ have: 1. δ(Γ) = dim (Λ (Γ)). H c In particular, if Γ is geometrically finite, Λ(Γ) Λ (Γ) is at most countable and we c \ obtain δ = dim (Λ (Γ)). Γ H c 2. If Γ is geometrically finite then either Λ(Γ) = Sn−1 or δ(Γ) < n 1. − Here dim is the Hausdorff dimension and Λ (Γ) Sn−1 is the conical limit set of Γ. H c ⊂ Thus the critical exponent of a Kleinian group is easy to estimate from above: δ(Γ) n 1. ≤ − Estimates from below, however, are nontrivial; our main theorem provides such a lower bound. 6 2.2 Algebraic preliminaries In this section we collect various definitions and results of homological algebra. We refer the reader to [7], [8] and [14] for the detailed discussion. For the rest of the paper, we let R be a commutative ring with a unit denoted 1. We note that although [8] and [14] restrict their discussion to R = Z, the definitions and facts that we will need directly generalize to the general commutative rings. Suggestion to the reader. For most of the paper, the reader uncomfortable with homological algebra can think of (co)homology of Γ with trivial coefficients and of existence of a finite K(Γ,1) instead of the finite type condition for Γ. However in the proofs of Theorem 1.3 and Corollary 1.5, we need (co)homology with twisted coefficients as well as the general notion of finite type. A group Γ is said to be of finite type, or FP (over R), if there exists a resolution by finitely generated projective RΓ–modules 0 P P ... P R 0. k k−1 0 → → → → → → For instance, if there exists a finite cell complex K = K(Γ,1), then Γ has finite type for every ring R. Every group of finite type is finitely generated, although it does not have to be finitely-presented, see [5]. More generally, a group Γ is said to be of type FP (over R), if there exists a k partial resolution by finitely generated projective RΓ–modules P P ... P R 0. k k−1 0 → → → → → A group Γ is said to have cohomological dimension k if k is the least integer such that there exists a resolution by projective RΓ–modules 0 P P ... P R 0. k k−1 0 → → → → → → Lemma 2.3. Suppose that Γ is of type FP and cd(Γ) k. Then Γ is of type FP. k ≤ Proof. See discussion following the proof of Proposition 6.1 in [14, Chapter VIII]. A group Γ is said to have homological (or weak) dimension k over R, if k is the least integer such that there exists a resolution by flat RΓ–modules 0 F F ... F R 0. k k−1 0 → → → → → → Thus the (co)homological dimension of Γ equals the (projective) flat dimension of the Γ–module RΓ. The cohomological and homological dimensions of Γ are denoted by cd (Γ) and hd (Γ) respectively. One can restate the definition of (co)homological R R dimension in terms of vanishing of (co)homologies of Γ: Theorem 2.4. (See [7].) cd (Γ) = sup n : an RΓ–module V so that Hn(Γ;V) = 0 , R { ∃ 6 } hd (Γ) = sup n : an RΓ–module V so that H (Γ;V) = 0 . R n { ∃ 6 } 7 Theorem 2.5. Let Γ be a torsion-free group such that cd (Γ) 1. Then Γ is free. R ≤ This theorem was originally proven by Stallings [24] for finitely-generated groups and R = Z; his proof was extended by Swan [26] to arbitrary groups. Finally, Dunwoody [16] proved this theorem for arbitrary rings. Remark 2.6. One can weaken the torsion-free assumption, by restricting to groups with torsion of bounded order, see [16]. We will need a generalization of these definitions to the relative case. In what follows we let Γ be a group and Π be a nonempty collection of subgroups Π := Π ,i I . i { ∈ } Given an RΓ-module V, one defines the relative (co)homology groups H (Γ,Π;V), H∗(Γ,Π;V). ∗ Instead of the algebraic definition of (co)homologies with coefficients in an RΓ- module V, we will be using the topological interpretation, following [8, Section 1.5]. Let K := K(Γ,1) be an Eilenberg-MacLane space for Γ. Let C := K(Π ,1), i I. i i ∈ We assume that the complexes C are embedded in K, so that C C = for i = j. i i j ∩ ∅ 6 Set C := C . i [ i∈I Then we will be computing the (co)homologies of the pair (Γ,Π) using the relative (co)homologies of (K,C). Namely, let X denote the universal cover of K. Let V be R the module V, regarded as an R–module. We obtain the trivial (product) sheaf V˜ over X with fibers V . We will think of this sheaf as the sheaf of local (horizontal) R sections of the product bundle E := X V X. By abusing the notation we R × → will identify bundles and sheafs of their sections. The group Γ acts on this sheaf diagonally: γ (x,v) = (γ(x),γ v),γ Γ. · · ∈ The bundle E (and the sheaf V˜) project to the space K, to a bundle V K and → its sheaf V of local horizontal sections. Then we have natural isomorphisms H∗(Γ,Π;V) = H∗(K,C;V), H (Γ,Π;V) = H (K,C;V). ∼ ∗ ∼ ∗ We will mostly work with the relative homology groups H (K,C;V), which we ∗ will think of as the (relative) singular homology of K (rel. C) with coefficients in V. We refer the reader to [20] for the precise definition. The most important example (for us) of this computation of relative homologies will be when Γ is a Kleinian group, the complex K is the hyperbolic manifold M = Hn/Γ, and the subcomplex C is a disjoint union of Margulis tubes and cusps in M. More generally, we will consider the case when K is a metrically complete connected hyperbolic manifold with convex boundary. We now return to the general case of group pairs (Γ,Π). 8 Definition 2.7. The relative (co)homological dimension of Γ (rel. Π) is defined as cd (Γ,Π) = sup n : an RΓ–module V so that Hn(Γ,Π;V) = 0 , R { ∃ 6 } hd (Γ,Π) = sup n : an RΓ–module V so that H (Γ,Π;V) = 0 . R n { ∃ 6 } In the case of R = Z, we will omit the subscript from the notation for the (co)homological dimension. Set RΓ/Π := RΓ/Π . i∈I i ⊕ We have the augmentation ǫ : RΓ/Π R, given by ǫ(gΠ ) := 1 for all cosets gΠ and i i → all i. Following [8, Section 1.1], we set ∆ := ∆ := Ker(ǫ). Γ/Π Then (see [8, Section 1.1]) Hk(Γ,Π;V) = Hk−1(Γ;Hom(∆,V)), ∼ H (Γ,Π;V) = H (Γ;∆ V). k ∼ k−1 ⊗ The cohomological and homological dimensions of (Γ,Π) can be interpreted as flat and projective dimensions of ∆ = ∆ respectively: Γ/Π hd (Γ,Π) 1 = flatdim(∆), cd (Γ,Π) 1 = projdim(∆), (2) R R − − see [8, Section 4.1]. For most of the paper this interpretation of (co)homological dimension will be unnecessary; the only exceptions are Lemmata 2.8 and 2.9 below: Lemma 2.8. hd (Γ,Π) cd (Γ,Π) hd (Γ,Π)+1. R R R ≤ ≤ Proof. The absolute case was proved in [7]; the relative case follows from the same arguments as in Bieri’s book using the equation (2). A pair (Γ,Π) is said to have finite type (over R) if: 1. Γ and each Π has type FP. i 2. The set I is finite. This condition is stronger than the one considered in [8, Section 4.1]. However it will suffice for our purposes as we are interested in the case where each Π is a finitely i generated virtually abelian group. Such groups Π necessarily have finite type. i Note that there is a free finitely generated Kleinian group Γ Isom(H4), so that ⊂ Γ contains infinitely many Γ-conjugacy classes of maximal parabolic subgroups, [22]. It is unknown if every Kleinian group Γ of finite type contains only finitely many conjugacy classes of maximal parabolic subgroups of rank 2. ≥ If Γ Isom(Hn) is a geometrically finite Kleinian group, then it contains only ⊂ finitely many conjugacy classes of maximal parabolic subgroups, see [11]. Moreover, Γ has finite type since its admits a finite K(Γ,1), which is the complement to cusps in the convex core of Hn/Γ. Therefore in this case (Γ,Π) has finite type. 9 Lemma 2.9. If (Γ,Π) is of finite type, then 1. cd (Γ,Π) = hd (Γ,Π). R R 2. cd (Γ,Π) = sup n : Hn(Γ,Π;RΓ) = 0 . R { 6 } Proof. This theorem was proved in [7] (see also [14, Chapter VIII, Proposition 6.7]) in the case when Π = . The same arguments go through in the relative case. ∅ SupposethatΓisvirtuallytorsionfree,i.e. itcontainsafinite-indexsubgroupΓ′ ⊂ Γ which is torsion-free. Let Π′ denote the collection of subgroups of Γ′ obtained by intersecting Γ′ withtheelements ofΠ. Onedefinesthevirtualrelative(co)homological dimension of Γ as vcd (Γ,Π) = cd (Γ′,Π′), R R vhd (Γ,Π) = hd (Γ′,Π′). R R Recall that every finitely-generated Kleinian group is virtually torsion-free by Sel- berg’s lemma. 3 Volumes of relative cycles Let X be either a simplicial complex or a Riemannian manifold, possibly with convex boundary. In the case when X is a simplicial complex, we metrize X by identifying each i-simplex in X with the standard Euclidean i-simplex in Ri+1. Let Y X be ⊂ either a subcomplex or a closed submanifold with piecewise-smooth boundary. Let ωˆ be the q-volume form on X induced by piecewise-Euclidean or Riemannian metric q on X. Let χ be the characteristic function of X Y; we define the relative q-volume \ form ω by q ω := χ ωˆ . q q · Let W X bea flat bundle whose fibers arecopies ofanR-moduleV . We define R → the relative volume Vol(ζ,Y) for piecewise-smooth singular q-chains ζ in C (X,W) q as follows. Consider first the case when ζ = w σ, where σ : ∆q X is a singular ⊗ → q-simplex and w is a (horizontal) section of W over the support of σ. Then set Vol(ζ,Y) = σ∗(ω ). q Z ∆q For a general chain s ζ = w σ , i i ⊗ Xi=1 set s Vol(ζ,Y) := Vol(w σ ). i i ⊗ Xi=1 We set Vol(ζ) := Vol(ζ, ). Clearly, the relative volume descends to a function on ∅ Z (X,Y;W). For a relative homology class ξ H (X,Y;W), we define the relative q q ∈ volume by Vol(ξ,Y) := inf Vol(ζ,Y) : ξ = [ζ] . { } 10