Torsion homology growth and cheap rebuilding of inner-amenable groups
arXiv:2212.07916 · doi:10.4171/GGD/803
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 -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.
17 pages; v2: added missing hypothesis in Cor. 6.8, added Lemma 2.2 and Examples 2.4, 2.5; v3: replaced 'torsion-free' by 'virtually torsion-free'; final version, to appear in Groups, Geometry and Dynamics