We consider Moebius and conformal homeomorphisms f : ∂X → ∂Y between boundaries of CAT(-1) spaces X, Y equipped with visual metrics. A conformal map f induces a topological conjugacy of the geodesic flows of X and Y , which is flip-equivariant if f is Moebius. We define a function S(f ) : ∂ 2 X → R, the integrated Schwarzian of f , which measures the deviation of the topological conjugacy from being flip-equivariant, in particular vanishing if f is Moebius. Conversely if X, Y are simply connected complete manifolds with pinched negative sectional curvatures, then f is Moebius on any open set U ⊂ ∂X such that S(f ) vanishes on ∂ 2 U . Indeed we obtain an explicit formula for the cross-ratio distortion in terms of the integrated Schwarzian. For such manifolds, we show that there is a Moebius homeomorphism f : ∂X → ∂Y if and only if there is a topological conjugacy of geodesic flows φ : T 1 X → T 1 Y with a certain uniform continuity property along geodesics.We show that if X, Y are proper, geodesically complete CAT(-1) spaces then any Moebius homeomorphism f extends to a (1, log 2)-quasi-isometry with image 1 2 log 2-dense in Y . We prove that if X, Y are in addition metric trees then f extends to a surjective isometry. The proofs involve a study of a space M(∂X) of metrics on ∂X Moebius equivalent to a visual metric and a natural isometric embedding of X into M(∂X). For C 1 conformal maps f : ∂X → ∂Y with bounded integrated Schwarzian and with domain X a simply connected negatively curved manifold with a lower bound on sectional curvature, similar arguments show that f extends to a (1, log 2 + 12||S(f )||∞) quasi-isometry.We also obtain a dynamical classification of Moebius self-maps f : ∂X → ∂X into three types, elliptic, parabolic and hyperbolic.
We consider a log-Riemann surface S with a finite number of ramification points and finitely generated fundamental group. The log-Riemann surface is equipped with a local holomorphic difffeomorphism π : S → C. We prove that S is biholomorphic to a compact Riemann surface with finitely many punctures S, and the pull-back of the 1-form dπ under the biholomorphic map φ : S → S is a 1-form ω = φ * dπ with isolated singularities at the punctures of exponential type, i.e. near each puncture p, ω = e h · ω 0 where h is a function meromorphic near p and ω 0 a 1-form meromorphic near p.
Abstract. For i = 1, 2, let G i be cocompact groups of isometries of hyperbolic space H n of real dimension n, n ≥ 3. Let H i ⊂ G i be infinite index quasiconvex subgroups satisfying one of the following conditions: 1) limit set of H i is a codimension one topological sphere. 2) limit set of H i is an even dimensional topological sphere.3) H i is a codimension one duality group. This generalizes (1). In particular, if n = 3, H i could be any freely indecomposable subgroup of G i . 4) H i is an odd-dimensional Poincaré Duality group P D(2k + 1). This generalizes (2). We prove pattern rigidity for such pairs extending work of Schwartz who proved pattern rigidity when H i is cyclic. All this generalizes to quasiconvex subgroups of uniform lattices in rank one symmetric spaces satisfying one of the conditions (1)- (4), as well as certain special subgroups with disconnected limit sets. In particular, pattern rigidity holds for all quasiconvex subgroups of hyperbolic 3-manifolds that are not virtually free. Combining this with the main result of Mosher-Sageev-Whyte [MSW04], we get quasi-isometric rigidity results for graphs of groups where the vertex groups are uniform lattices in rank one symmetric spaces and edge groups are of any of the above types.This paper is dedicated to the memory of Kalyan Mukherjea.
We present a construction of hedgehogs for holomorphic maps with an indifferent fixed point. We construct, for a family of commuting non-linearisable maps, a common hedgehog of Hausdorff dimension 1, the minimum possible.
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