Math. Z. (2008) 259:249–253 DOI 10.1007/s00209-007-0221-5
Mathematische Zeitschrift
A Bowen type rigidity theorem for non-cocompact hyperbolic groups Xiangdong Xie
Received: 18 March 2007 / Accepted: 14 June 2007 / Published online: 21 July 2007 © Springer-Verlag 2007
Abstract We establish a Bowen type rigidity theorem for the fundamental group of a noncompact hyperbolic manifold of finite volume (with dimension at least 3). Keywords Rigidity theorem · Geometrically finite group · Hausdorff dimension · Global measure formula Mathematics Subject Classification (2000) 53C24 · 30F40 · 37F30
1 Introduction A classical theorem of Bowen [4] says that if the fundamental group of a closed hyperbolic surface acts convex-cocompactly on H3 , then the limit set has Hausdorff dimension at least 1, and equality holds only when the limit set is a round circle. Here we are using the ball model for H3 . This result was later generalized to higher dimensions by Yue [13], and to the CAT(−1) setting by Bonk-Kleiner [6]. We generalize this result in another direction: we consider noncompact finite volume hyperbolic manifolds (with dimension ≥3) and geometrically finite actions of their fundamental groups on real hyperbolic spaces. See Theorem 3.1. Notice that a convex-cocompact group is a geometrically finite group without parabolic elements. Our result should also hold for other rank one symmetric spaces. The proof here also applies in those cases, provided one verifies Tukia’s homeomorphism theorem [12] for other rank one symmetric spaces and Stratmann-Velani’s Global Measure Formula [11] for quarternionic hyperbolic spaces and the Cayley plane. The Global Measure Formula has been established by Newberger [8] for geometrically finite complex hyperbolic groups.
X. Xie (B) Department of Mathematics, Virginia Tech, Blacksburg, VA 24060-0123, USA e-mail:
[email protected]
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2 Preliminaries Let Bm+1 ⊂ Rm+1 be the unit ball with the hyperbolic metric, Sm = ∂ Bm+1 ⊂ Rm+1 the unit sphere with the Euclidean metric (the restriction of the Euclidean metric on Rm+1 ). Let G ⊂ Isom(Bm+1 ) be a torsion-free geometrically finite group, (G) ⊂ Sm the limit set of G and (G) := Sm \(G) the set of discontinuity. Denote by δ(G) the critical exponent of G, and µ the Patterson measure based at the origin o. Recall that the manifold (possibly with boundary) M = Bm+1 ∪(G) /G can be written as a disjoint union of a compact set C and a finite number of cusp ends Ci : M =C∪ Ci . See [3, p. 272]. The fundamental group of each cusp end is a maximal parabolic subgroup of G. We fix such a decomposition for M.
Stratmann-Velani’s Global Measure Formula Let G ⊂ Isom(Bm+1 ) be a torsion-free geometrically finite group. Recall that each parabolic subgroup has a finite index subgroup that is free abelian. The rank of a parabolic subgroup is defined to be the rank of a finite index free abelian subgroup. Notice that this is well-defined. Let ξ ∈ (G) and t > 0. Denote by ξt the point on the ray oξ at distance t from o. Define k(ξt ) to be δ(G) if the projection of ξt in M lies in C, and to be the rank of π1 (Ci ) if the projection of ξt in M lies in Ci . Let Bt (ξ ) ⊂ Sm be the open ball with center ξ and radius e−t , and (ξt ) := d(ξt , G(o)) the distance from ξt to the orbit of o. Theorem 2.1 (Global Measure Formula [11]) There is a constant C ≥ 1 with the following property: for all ξ ∈ (G) and t > 0, we have 1 µ(Bt (ξ )) ≤ −tδ(G)−(ξ )(δ(G)−k(ξ )) ≤ C. t t C e Tukia’s Homeomorphism Theorem Let η : [0, ∞) → [0, ∞) be a homeomorphism. A homeomorphism between metric spaces f : X → Y is η-quasisymmetric if for all distinct triples x, y, z ∈ X , we have d(x, y) d( f (x), f (y)) ≤η . d( f (x), f (z)) d(x, z) A homeomorphism f : X → Y is a quasisymmetric map if it is η-quasisymmetric for some η. Theorem 2.2 ([12] Theorem 3.3) For i = 1, 2, let G i ⊂ Isom(Bm i +1 ) be a geometrically finite group. If φ : G 1 → G 2 is a type-preserving isomorphism, then there is an equivariant quasisymmetric map F : (G 1 ) → (G 2 ). Recall that φ : G 1 → G 2 is type-preserving means g ∈ G 1 is a parabolic element if and only if φ(g) is a parabolic element. The map F : (G 1 ) → (G 2 ) is equivariant if F(g(ξ )) = φ(g)(F(ξ )) for all g ∈ G 1 and all ξ ∈ (G 1 ).
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Pansu’s Lemma A Borel regular measure m on a metric space X is Ahlfors regular if there exist constants α > 0 and C ≥ 1 such that the following holds 1 α r ≤ m(B(x, r )) ≤ Cr α C for all x ∈ X and all 0 < r < diam(X ), where B(x, r ) is the open ball with center x and radius r . See [7, p. 62]. In this case, α agrees with the Hausdorff dimension Hdim(X ) of X and the Hausdorff measure is comparable with m. In particular, the Hausdorff measure is also Ahlfors regular. We say a metric space X is Ahlfors regular if 0 < Hdim(X ) < ∞ and its Hausdorff measure is Ahlfors regular. Notice that a Borel regular measure on a Ahlfors regular metric space does not have to be Ahlfors regular. For any metric space X , the Ahlfors regular conformal dimension of X is defined as Cdim(X ) = inf{Hdim(Y )}, where the infimum is taken over all Ahlfors regular metric spaces Y that are quasisymmetric to X . Lemma 2.3(a) follows from the proof of Proposition 6.5 in [9] and Lemma 2.3(b) is a consequence of Corollary 7.2 in [9]. Lemma 2.3 Let E be an Ahlfors regular metric space with Hdim(E) > 1. Suppose there is a family of curves F in E equipped with a measure ν satisfying the following property: there is a constant C ≥ 1 such that for every ball B(x, r ) ⊂ E with r ≤ diam(E) we have C −1 r Hdim(E)−1 ≤ ν({γ ∈ F : γ ∩ B(x, r ) = ∅}) ≤ Cr Hdim(E)−1 . Then (a) Hdim(E) = Cdim(E); (b) The Hausdorff measure m of E is minimal in the following sense: Let Y be an Ahlfors regular metric space with Hausdorff measure m and f : E → Y a quasisymmetric map. If Hdim(Y ) = Cdim(E), then m is absolutely continuous with respect to f ∗ m . It was proved in [9, p. 21–22] that the ideal boundary (equipped with the Carnot metric) of rank one symmetric spaces with dimension > 2 satisfies the assumption in the above lemma. In particular, it is the case with Sm (m ≥ 2).
3 Proof of the main theorem Now we are ready to prove the main theorem. The restriction on dimension is due to Pansu’s lemma (see Lemma 3.2 and the remark after it). Theorem 3.1 Let G 1 ⊂ Isom(Bn+1 ), G 2 ⊂ Isom(B N +1 ) (n, N ≥ 2) be torsion-free geometrically finite groups. Assume Bn+1 /G 1 is noncompact and has finite volume. Let φ : G 1 → G 2 be a type-preserving isomorphism. Then Hdim((G 2 )) ≥ n. Furthermore, equality holds if and only if G 2 stabilizes an isometric copy of Bn+1 in B N +1 . Let µi (i = 1, 2) be the Patterson measure of G i based at o. Lemma 3.2 Hdim((G 2 )) ≥ n. If equality holds, then µ2 is Ahlfors regular with dimension n.
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Proof By Tukia’s theorem there is an equivariant quasisymmetric map F : (G 1 ) → (G 2 ). In particular, (G 2 ) is homeomorphic to (G 1 ) = Sn . Since Hausdorff dimension is bounded below by topological dimension, we have Hdim((G 2 )) ≥ dim((G 2 )) = dim(Sn ) = n. Here dim denotes topological dimension. Suppose Hdim((G 2 )) = n. Recall that (G 2 ) consists of radial limit points and a countable number of parabolic points. It follows from the theorem of Bishop-Jones [5] that δ(G 2 ) = Hdim((G 2 )). Hence δ(G 2 ) = n. On the other hand, since Bn+1 /G 1 has finite volume, all maximal parabolic subgroups of G 1 have rank n. Since φ : G 1 → G 2 is typepreserving, all maximal parabolic subgroups of G 2 also have rank n. It follows that k(ξt ) = n for all ξ ∈ (G 2 ) and all t. Now the Global Measure Formula implies that µ2 is Ahlfors regular with dimension n.
It follows from Lemma 3.2 that µ2 is comparable with the Hausdorff measure if Hdim((G 2 )) = n. Notice that the Global Measure Formula implies that µ1 is Ahlfors regular with dimension n and hence is also comparable with the Hausdorff measure. Let X be a metric space and Q = (x1 , x2 , x3 , x4 ) be a quadruple of distinct points in X . The cross ratio of Q is cr (Q) =
d(x1 , x3 )d(x2 , x4 ) . d(x1 , x4 )d(x2 , x3 )
An embedding f : X → Y of metric spaces is called a Möbius embedding if cross ratio is preserved, that is, cr ( f (Q)) = cr (Q) for all quadruples of distinct points in X . For any CAT(−1) metric space, Bourdon ([1], Sect. 2.5) defined a family of visual metrics on the ideal boundary. He proved that (Theorem 0.1. of [2]) if X is a rank one symmetric space and Y is a CAT(−1) space, then every Möbius embedding g : ∂ X → ∂Y is induced by an isometric embedding, that is, there is an isometric embedding G : X → Y such that the boundary map of G agrees with g. Here ∂ X and ∂Y are equipped with Bourdon’s metric. In the case of Bm+1 and base point o, Bourdon’s metric on Sm is half the Euclidean metric. The following argument is due to Sullivan [10]. We include it here only for completeness. See also [2]. Completing the proof of Theorem 3.1 Let i2 = {(ξ, η) ∈ (G i ) × (G i ) : ξ = η}. Let νi be the measure on i2 defined by νi (ξ, η) =
µi (ξ ) × µi (η) . (d(ξ, η))2n
Then νi is invariant under the diagonal action of G i . Since G i is geometrically finite, the diagonal action of G i on i2 is ergodic with respect to νi . See for example Theorem A in [14]. Suppose Hdim((G 2 )) = n. We have observed that µi is comparable with the Hausdorff measure. Now Lemma 3.2 and Pansu’s lemma imply that µ1 is absolutely continuous with respect to F ∗ µ2 . Hence µ1 (ξ ) = h(ξ )F ∗ µ2 (ξ ) for some F ∗ µ2 measurable function h. Now we have F ∗ µ2 (ξ ) × F ∗ µ2 (η) ν1 (ξ, η) = h(ξ )h(η) (d(ξ, η))2n and (F × F)∗ ν2 (ξ, η) =
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F ∗ µ2 (ξ ) × F ∗ µ2 (η) . (d(F(ξ ), F(η)))2n
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Since F : (G 1 ) → (G 2 ) is equivariant, the ergodicity implies that (F × F)∗ ν2 = Cν1 for some constant C > 0. Hence the following holds F ∗ µ2 × F ∗ µ2 -a.e., d(ξ, η) = C 1/2n h 1/2n (ξ )h 1/2n (η)d(F(ξ ), F(η)). Now the continuity of F implies that F is a Möbius embedding. Theorem 3.1 then follows from Bourdon’s theorem (Theorem 0.1. of [2]). Acknowledgments The author particularly thanks the Department of Mathematics at Virginia Tech for its generous support: the teaching load of one class per year is really a gift.
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