Tukia's isomorphism theorem in CAT(-1) spaces

TL;DR

This paper generalizes Tukia's isomorphism theorem from hyperbolic spaces to CAT(-1) spaces, identifying conditions under which boundary extensions are quasisymmetric and extending rigidity results to symmetric spaces of noncompact type.

Contribution

It extends Tukia's theorem to CAT(-1) spaces, providing conditions for boundary extension as quasisymmetric maps and generalizing Xie's rigidity theorem to finite-dimensional symmetric spaces.

Findings

Boundary extensions are not always quasisymmetric in CAT(-1) spaces.

A lattice and matching base fields ensure quasisymmetric boundary extension.

Generalization of Xie's rigidity theorem to finite-dimensional symmetric spaces.

Abstract

We prove a generalization of Tukia's ('85) isomorphism theorem which states that isomorphisms between geometrically finite groups extend equivariantly to the boundary. Tukia worked in the setting of real hyperbolic spaces of finite dimension, and his theorem cannot be generalized as stated to the setting of CAT($-1$) spaces. We exhibit examples of type-preserving isomorphisms of geometrically finite subgroups of finite-dimensional rank one symmetric spaces of noncompact type (ROSSONCTs) whose boundary extensions are not quasisymmetric. A sufficient condition for a type-preserving isomorphism to extend to a quasisymmetric equivariant homeomorphism between limit sets is that one of the groups in question is a lattice, and that the underlying base fields are the same, or if they are not the same then the base field of the space on which the lattice acts has the larger dimension. This in turn leads to a generalization of a rigidity theorem of Xie ('08) to the setting of finite-dimensional ROSSONCTs.