Ultrametric analogues of Ulam stability of groups

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 (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: these include finite groups, virtually free groups, some groups acting on rooted trees, and certain lamplighter and (Generalized) Baumslag-Solitar groups.

Asymptotic Cohomology and Uniform Stability for Lattices in Semisimple Groups

“…A recent monograph of [FF21] studies stability with respect to p-adic groups. An interesting feature in this non-archimedean setting is that the ultrametric (strong triangle inequality) forces an equivalence between uniform and pointwise stability (for finitely presented groups).…”