On the bounded cohomology of semi-simple groups, S-arithmetic groups and products

TL;DR

This paper proves vanishing results in bounded cohomology for semi-simple, S-arithmetic, and product groups, revealing rigidity properties and establishing bounds related to the groups' ranks and structures.

Contribution

It introduces new vanishing theorems in bounded cohomology for a broad class of groups, including semi-simple and S-arithmetic groups, and explores their rigidity and product structures.

Findings

Vanishing of bounded cohomology below twice the rank for semi-simple groups.

Rigidity results for S-arithmetic groups and groups over global fields.

Vanishing and rigidity results for products of locally compact groups and their lattices.

Abstract

We prove vanishing results for Lie groups and algebraic groups (over any local field) in bounded cohomology. The main result is a vanishing below twice the rank for semi-simple groups. Related rigidity results are established for S-arithmetic groups and groups over global fields. We also establish vanishing and cohomological rigidity results for products of general locally compact groups and their lattices.