On growth of homology torsion in amenable groups

TL;DR

This paper investigates the growth of torsion in homology groups of certain spaces under group actions, showing subexponential growth in specific cases and contrasting with faster growth in non-compact scenarios.

Contribution

It establishes subexponential growth of homology torsion in amenable groups acting on simply connected complexes with compact quotient, and contrasts with cases where the quotient is non-compact.

Findings

Torsion growth is subexponential for amenable groups with compact quotient.

Non-compact quotients can have torsion growth faster than any given function.

Provides examples of solvable groups with rapid torsion growth.

Abstract

Suppose an amenable group $G$ is acting freely on a simply connected simplicial complex $\tilde X$ with compact quotient $X$. Fix $n \geq 1$, assume $H_n(\tilde X, \mathbb{Z})=0$ and let $(H_i)$ be a Farber chain in $G$. We prove that the torsion of the integral homology in dimension $n$ of $\tilde{X}/H_i$ grows subexponentially in $[G:H_i]$. By way of contrast, if $X$ is not compact, there are solvable groups of derived length 3 for which torsion in homology can grow faster than any given function.