Lamplighters over non-amenable groups are not strongly Ulam stable

TL;DR

This paper demonstrates that wreath products of abelian and non-amenable groups lack strong Ulam stability and deformation rigidity, extending previous results to a broader class of groups.

Contribution

It establishes non-strong Ulam stability and non-deformation rigidity for wreath products of abelian and non-amenable groups, generalizing prior findings.

Findings

Wreath products of abelian and non-amenable groups are not strongly Ulam stable.

These wreath products are not deformation rigid.

Extends known results to a wider class of non-amenable group constructions.

Abstract

We prove that a wreath product of an abelian group and a non-amenable group is not strongly Ulam stable. Previously this was known for groups containing free subgroups, due to work of Burger, Ozawa and Thom, and for some surface groups, due to work by Kazhdan. We also show that these wreath products are not deformation rigid.