Ultrametric analogues of Ulam stability of groups

TL;DR

This paper investigates the stability of metric group approximations using ultrametrics, especially in the p-adic setting, establishing connections with residual finiteness and providing examples and criteria for stability.

Contribution

It introduces ultrametric analogues of Ulam stability, shows their equivalence with residual finiteness, and offers new examples and a cohomological criterion for stability.

Findings

Ultrametric stability is equivalent to residual finiteness for finitely presented groups.

Several classes of groups are shown to be uniformly stable, including finite and virtually free groups.

A finitely generated non-uniformly stable group is constructed.

Abstract

We study stability of metric approximations of countable groups with respect to groups endowed with ultrametrics, the main case study being a $p$-adic analogue of Ulam stability, where we take $GL_n(\mathbb{Z}_p)$ as approximating groups instead of $U(n)$. For finitely presented groups, the ultrametric nature implies equivalence of the pointwise and uniform stability problems, and the profinite one implies that the corresponding approximation property is equivalent to residual finiteness. Moreover, a group is uniformly stable if and only if its largest residually finite quotient is. We provide several examples of uniformly stable groups, including finite groups, virtually free groups, some groups acting on rooted trees, and certain lamplighter and (Generalised) Baumslag--Solitar groups. We construct a finitely generated group that is not uniformly stable. Finally, we prove and apply a (bounded) cohomological criterion for stability of a finitely presented group.