Torsion homology growth and cheap rebuilding of 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 Abért, Bergeron, Frączyk 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.