Flows, Fixed Points and Rigidity for Kleinian Groups

TL;DR

This paper investigates the rigidity properties of Kleinian groups and their boundary homeomorphisms, establishing new results that extend Mostow rigidity to more general settings involving patterns and quasiconformal maps.

Contribution

It introduces a relative version of Mostow rigidity called pattern rigidity, generalizing previous results to broader classes of invariant patterns under Kleinian groups.

Findings

Non-trivial one-parameter subgroups arise when conjugacy is not conformal.

Quasiconformal maps pairing invariant patterns are necessarily conformal in higher dimensions.

Generalizes previous rigidity results for limit sets and Poincaré duality subgroups.

Abstract

We study the closed group of homeomorphisms of the boundary of real hyperbolic space generated by a cocompact Kleinian group $G_1$ and a quasiconformal conjugate $h^{-1}G_2 h$ of a cocompact group $G_2$. We show that if the conjugacy $h$ is not conformal then this group contains a non-trivial one parameter subgroup. This leads to rigidity results; for example, Mostow rigidity is an immediate consequence. We are also able to prove a relative version of Mostow rigidity, called pattern rigidity. For a cocompact group $G$, by a $G$-invariant pattern we mean a $G$-invariant collection of closed proper subsets of the boundary of hyperbolic space which is discrete in the space of compact subsets minus singletons. Such a pattern arises for example as the collection of translates of limit sets of finitely many infinite index quasiconvex subgroups of $G$. We prove that (in dimension at least three) for $G_1, G_2$ cocompact Kleinian groups, any quasiconformal map pairing a $G_1$-invariant pattern to a $G_2$-invariant pattern must be conformal. This generalizes a previous result of Schwartz who proved rigidity in the case of limit sets of cyclic subgroups, and Biswas-Mj who proved rigidity for Poincare Duality subgroups.