A note on bounded-cohomological dimension of discrete groups

TL;DR

This paper explores the properties of bounded-cohomological dimension in groups, providing new examples of groups with zero dimension and methods to embed any group into acyclic groups with trivial bounded cohomology.

Contribution

It introduces constructions for groups with infinite bounded-cohomological dimension and demonstrates that every group can embed into an acyclic group with trivial bounded cohomology.

Findings

Existence of groups with infinite bounded-cohomological dimension

Construction of groups with bounded-cohomological dimension zero

Embedding of any group into acyclic groups with trivial bounded cohomology

Abstract

Bounded-cohomological dimension of groups is a relative of classical cohomological dimension, defined in terms of bounded cohomology with trivial coefficients instead of ordinary group cohomology. We will discuss constructions that lead to groups with infinite bounded-cohomological dimension, and we will provide new examples of groups with bounded-cohomological dimension equal to 0. In particular, we will prove that every group functorially embeds into an acyclic group with trivial bounded cohomology.