Torsion homology growth and cheap rebuilding of inner-amenable groups

TL;DR

This paper demonstrates that certain inner-amenable, non-amenable groups have the cheap 1-rebuilding property, leading to vanishing first $$-Betti numbers and log-torsion, extending known results from amenable groups.

Contribution

It extends the understanding of torsion homology growth to inner-amenable groups using a new structural approach and the concept of cheap 1-rebuilding.

Findings

Vanishing first $$-Betti number for these groups

Log-torsion in degree 1 vanishes

Extension of results from amenable to inner-amenable groups

Abstract

We prove that virtually torsion-free, residually finite groups that are inner-amenable and non-amenable have the cheap 1-rebuilding property, a notion recently introduced by Ab\'ert, Bergeron, Fr\k{a}czyk and Gaboriau. As a consequence, the first $\ell^2$-Betti number with arbitrary field coefficients and log-torsion in degree 1 vanish for these groups. This extends results previously known for amenable groups to inner-amenable groups. We use a structure theorem of Tucker-Drob for inner-amenable groups showing the existence of a chain of q-normal subgroups.