Measurable Rigidity for Kleinian groups

TL;DR

This paper demonstrates the existence of a measurable boundary map between Kleinian groups with homeomorphic quotients, constructed explicitly from Cannon-Thurston maps, extending measurable rigidity results.

Contribution

It constructs explicitly a measurable boundary map for divergence type Kleinian groups using Cannon-Thurston maps, confirming a key aspect of measurable rigidity.

Findings

Existence of a measurable boundary map between Kleinian groups with homeomorphic quotients.

Construction of the boundary map explicitly from Cannon-Thurston maps.

Extension of measurable rigidity theorems to broader classes of Kleinian groups.

Abstract

Let $G, H$ be two Kleinian groups with homeomorphic quotients $\mathbb H^3/G$ and $\mathbb H^3/H$. We assume that $G$ is of divergence type, and consider the Patterson-Sullivan measures of $G$ and $H$. The measurable rigidity theorem by Sullivan and Tukia says that a measurable and essentially directly measurable equivariant boundary map $\widehat k$ from the limit set $\Lambda_G$ of $G$ to that of $H$ is either the restriction of a M\"{o}bius transformation or totally singular. In this paper, we shall show that such $\widehat k$ always exists. In fact, we shall construct $\widehat k$ concretely from the Cannon-Thurston maps of $G$ and $H$.