The algebraic cheap rebuilding property

TL;DR

This paper develops an axiomatic framework for combination theorems concerning various homological properties of groups and chain complexes, including an algebraic version of the cheap rebuilding property that impacts homology growth.

Contribution

It introduces an algebraic version of the cheap rebuilding property and integrates it into an axiomatic approach for homological properties of groups.

Findings

Graphs of groups with amenable vertex and edge groups have vanishing torsion homology growth.

The framework unifies multiple homological properties under a common axiomatic approach.

The algebraic cheap rebuilding property implies vanishing of torsion homology growth.

Abstract

We present an axiomatic approach to combination theorems for various homological properties of groups and, more generally, of chain complexes. Examples of such properties include algebraic finiteness properties, $\ell^2$-invisibility, $\ell^2$-acyclicity, lower bounds for Novikov--Shubin invariants, and vanishing of homology growth. As a key example, we introduce an algebraic version of Ab\'ert--Bergeron--Fr\k{a}czyk--Gaboriau's cheap rebuilding property that implies vanishing of torsion homology growth and fits into our axiomatic framework for combination theorems. In particular, we obtain that certain graphs of groups with amenable vertex groups and elementary amenable edge groups have vanishing torsion homology growth.