Isometry groups of proper CAT(0)-spaces

TL;DR

This paper investigates the structure of isometry groups of proper CAT(0)-spaces, showing that non-elementary groups with rank-one elements have rich cohomological properties and are closely related to either totally disconnected groups or rank-one Lie groups.

Contribution

It establishes a link between the presence of rank-one elements in isometry groups and their algebraic and cohomological structure, providing new classification insights.

Findings

Non-elementary groups with rank-one elements have non-trivial second bounded cohomology.

Such groups are essentially extensions of either totally disconnected groups or rank-one simple Lie groups.

The results connect geometric properties of CAT(0)-spaces with algebraic and cohomological group structures.

Abstract

Let G be a closed subgroup of the isometry group of a proper CAT(0)-space X. We show that if G is non-elementary and contains a rank-one element then its second bounded cohomology group with coefficients in the regular representation is non-trivial. As a consequence, up to passing to an open subgroup of finite index, either G is a compact extension of a totally disconnected group or G is a compact extension of a simple Lie group of rank one.