On Moebius and conformal maps between boundaries of {\rm CAT}(-1) spaces

Abstract

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.

Cross ratios on ${\rm CAT(0)}$ cube complexes and marked length-spectrum rigidity

“…On the other hand, no analogue of Theorem E is available even in CAT(−1) spaces and this is perhaps the most innovative feature of our work. We stress that the lack of an analogue of Theorem E for CAT(−1) Riemannian manifolds is the main reason why the marked length-spectrum rigidity conjecture (Problem 3.1 in [BK85]) is still wide open (see Theorem 5.1 in [Bis15]).…”

Quasi-metric antipodal spaces and maximal Gromov hyperbolic spaces

“…It is easy to see from the definition that the Busemann cocycle satisfies the cocycle identity B(x, z, ξ) = B(x, y, ξ) + B(y, z, ξ) for all x, y, z ∈ X, ξ ∈ ∂X. The following Lemma is well-known for CAT(-1) spaces (see [Bou96], [Bis15]), exactly the same proof goes through for boundary continuous spaces.…”

“…As in [Bis15], we have the following: Proposition 5.2. Let X be a proper, geodesically complete, boundary continuous Gromov hyperbolic space.…”

“…The converse, whether any Moebius map between boundaries f : ∂X → ∂Y extends to an isometry F : X → Y , is an open question. This is an important question: indeed, as explained in [Bis15], a positive answer to this question would yield a positive answer to the longstanding "marked length spectrum rigidity" conjecture of Burns and Katok ( [BK85]).…”

“…This is a natural condition to require when attempting to recover a space from its boundary. In [Bis15] it was shown that if X, Y are proper, geodesically complete CAT(-1) spaces, then any Moebius homeomorphism f : ∂X → ∂Y extends to a (1, log 2)-quasi-isometry F : X → Y . It was also shown that if X, Y are proper, geodesically complete trees, then any Moebius homeomorphism f : ∂X → ∂Y extends to a surjective isometry F : X → Y (see also [BS17] and [Hug04]), thus for trees and their boundaries we have a complete equivalence between isometries and Moebius maps.…”