A Dual Interpretation of the Gromov--Thurston Proof of Mostow Rigidity and Volume Rigidity for Representations of Hyperbolic Lattices

TL;DR

This paper provides a new proof of volume rigidity for hyperbolic lattice representations using bounded cohomology, offering a dual perspective to the classical Gromov–Thurston approach and extending Mostow Rigidity results.

Contribution

It introduces a bounded cohomology-based method to define volume and proves volume rigidity, offering a dual interpretation of Gromov–Thurston's proof of Mostow Rigidity.

Findings

Established a volume rigidity theorem for SO(n,1)-valued representations.

Provided a dual proof of Thurston's version of Gromov's proof of Mostow Rigidity.

Extended the rigidity results to non-cocompact lattices.

Abstract

We use bounded cohomology to define a notion of volume of an SO(n,1)-valued representation of a lattice SO(n,1) and, using this tool, we give a complete proof of the volume rigidity theorem of Francaviglia and Klaff in this setting. Our approach gives in particular a proof of Thurston's version of Gromov's proof of Mostow Rigidity (also in the non-cocompact case), which is dual to the Gromov--Thurston proof using the simplicial volume invariant.}