Asymptotic properties of groups acting on complexes

TL;DR

This paper investigates how asymptotic dimension and property A are preserved in groups acting on complexes, extending known results from graphs of groups to more general complexes.

Contribution

It proves that the fundamental group of a finite, developable complex of groups preserves finite asymptotic dimension and property A under certain conditions, extending previous graph-based results.

Findings

Finite asymptotic dimension is preserved under specific conditions.

Property A is preserved when the geometric realization has finite asymptotic dimension.

An example shows the necessity of the finite asymptotic dimension condition.

Abstract

We examine asymptotic dimension and property A for groups acting on complexes. In particular, we prove that the fundamental group of a finite, developable complex of groups will have finite asymptotic dimension provided the geometric realization of the development has finite asymptotic dimension and the vertex groups are finitely generated and have finite asymptotic dimension. We also prove that property A is preserved by this construction provided the geometric realization of the development has finite asymptotic dimension and the vertex groups all have property A. These results naturally extend the corresponding results on preservation of these large-scale properties for fundamental groups of graphs of groups. We also use an example to show that the requirement that the development have finite asymptotic dimension cannot be relaxed.