Shalom's property $H_{\mathrm{FD}}$ and extensions by $\mathbb{Z}$ of locally finite groups

Abstract

We show that every finitely generated extension by $\mathbb{Z}$ of a locally normally finite group has Shalom's property $H_{\mathrm{FD}}$. This is no longer true without the normality assumption. This permits to answer some questions of Shalom, Erschler-Ozawa and Kozma. We also obtain a Neumann-Neumann embedding result that any countable locally finite group embedds into a two generated amenable group with property $H_{\mathrm{FD}}$.

Added subsection 4.1, showing that lamplighter groups Z/2Z \wr Z^d for d at least 3 do not have property H_{FD}

References in corpus (1)

• Minimal growth harmonic functions on lamplighter groups