Bounded Cohomology and Unitary Representations of Automorphism Groups of Regular Trees

TL;DR

This paper computes the continuous bounded cohomology of automorphism groups of regular trees across all positive degrees for various irreducible unitary representations, revealing new non-zero cases.

Contribution

It provides the first comprehensive calculation of bounded cohomology for these groups with all irreducible unitary coefficients, filling a significant gap in the literature.

Findings

Bounded cohomology is non-zero in some cases for all positive degrees.

First complete determination of bounded cohomology for automorphism groups of regular trees.

Results apply to any irreducible continuous unitary representation.

Abstract

We compute the continuous bounded cohomology of the full automorphism groups of regular trees in all positive degrees, with coefficients arising from any irreducible continuous unitary representations. To the author's knowledge, this seems to be the first instance where the continuous bounded cohomology is determined in all positive degrees with coefficients arising from any irreducible continuous unitary representations without being zero in all cases.