Superrigidit\'{e} g\'{e}om\'{e}trique et applications harmoniques

TL;DR

This paper discusses the harmonic map approach to Margulis' superrigidity and arithmeticity theorems, and explores potential generalizations to fundamental groups of simplicial complexes with large spectral gaps, including random groups.

Contribution

It extends the harmonic map method to new classes of groups, such as those arising from simplicial complexes with spectral gap conditions, broadening the scope of superrigidity applications.

Findings

Harmonic map proof of Margulis' theorems explained

Potential generalizations to groups from simplicial complexes

Implications for random groups and spectral gaps

Abstract

These are expanded notes of a course given in Grenoble in june 2004. After a brief description of the harmonic map proof of Margulis' superrigidity and arithmeticity theorems, it is shown how the method might generalize to fundamental groups of simplicial complexes whose links have large enough nonlinear spectral gaps, with emphasis on random groups.