Integrable measure equivalence and rigidity of hyperbolic lattices

TL;DR

This paper explores the rigidity properties of hyperbolic lattices and surface groups through the lens of integrable measure equivalence, extending classical rigidity results and providing new classification insights.

Contribution

It generalizes Mostow rigidity to higher dimensions and introduces an integrable measure equivalence classification for hyperbolic lattices and surface groups.

Findings

Generalized Mostow rigidity for hyperbolic n-space with n>2

Cocycle version of strong rigidity established

Classification of lattices via integrable measure equivalence

Abstract

We study rigidity properties of lattices in hyperbolic n-space with n>2 and of surface groups in the context of (integrable) measure equivalence. The results for lattices in hyperbolic n-space with n>2 are generalizations of Mostow rigidity; they include a cocycle version of strong rigidity and an integrable measure equivalence classification. Despite the lack of Mostow rigidity for n=2 we show that cocompact lattices in SL(2,R) allow a similar integrable measure equivalence classification.