Ulam stability of lamplighters and Thompson groups

TL;DR

This paper demonstrates that many groups, including lamplighters and Thompson groups, are uniformly stable with respect to various unitary norms, using asymptotic cohomology techniques.

Contribution

It introduces new results on the uniform stability of lamplighter and Thompson groups using asymptotic cohomology, and explores hereditary and metric approximation properties.

Findings

Lamplighter groups are uniformly stable with respect to multiple norms.

Thompson groups exhibit uniform stability, extending previous understanding.

Foundational results in asymptotic cohomology are established.

Abstract

We show that a large family of groups is uniformly stable relative to unitary groups equipped with submultiplicative norms, such as the operator, Frobenius, and Schatten $p$-norms. These include lamplighters $\Gamma \wr \Lambda$ where $\Lambda$ is infinite and amenable, as well as several groups of dynamical origin such as the classical Thompson groups $F, F', T$ and $V$. We prove this by means of vanishing results in asymptotic cohomology, a theory introduced by the second author, Glebsky, Lubotzky and Monod, which is suitable for studying uniform stability. Along the way, we prove some foundational results in asymptotic cohomology, and use them to prove some hereditary features of Ulam stability. We further discuss metric approximation properties of such groups, taking values in unitary or symmetric groups.