Shalom’s property HFD and extensions by ℤ of locally finite groups

Abstract

We show that every finitely generated extension by ℤ of a locally normally finite group has Shalom’s property HFD. The statement 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 embeds into a two-generated amenable group with property HFD.