Torsion invariants of complexes of groups

TL;DR

This paper computes the homology torsion growth for groups acting on complexes, relating it to stabilizer subgroups and boundary topology, with applications to right-angled Artin groups and Lück approximation.

Contribution

It provides explicit formulas for torsion growth and L^2-torsion in complexes of groups, extending known results to new classes like right-angled Artin groups.

Findings

Homology torsion growth limits to boundary torsion

Formulas for mod p-homology growth under milder conditions

Complete torsion formulas for right-angled Artin groups

Abstract

Suppose a residually finite group $G$ acts cocompactly on a contractible complex with strict fundamental domain $Q$, where the stabilizers are either trivial or have normal $\mathbb{Z}$-subgroups. Let $\partial Q$ be the subcomplex of $Q$ with nontrivial stabilizers. Our main result is a computation of the homology torsion growth of a chain of finite index normal subgroups of $G$. We show that independent of the chain, the normalized torsion limits to the torsion of $\partial Q$, shifted a degree. Under milder assumptions of acyclicity of nontrivial stabilizers, we show similar formulas for the mod p-homology growth. We also obtain formulas for the universal and the usual $L^2$-torsion of $G$ in terms of the torsion of stabilizers and topology of $\partial Q$. In particular, we get complete answers for right-angled Artin groups, which shows they satisfy a torsion analogue of the L\"uck approximation theorem.