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 $G{L}_n({\mathbb{Z}}_p)$ 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.

arXiv comment90 pages. v2: added Section 8 with non-examples, improved the (bounded) cohomological criterion, developed the quantitative aspects. v3: minor changes. v4: accepted version. v5: small improvement in Theorem 1.10, final version for publication in Bulletin de la SMF

References in corpus (7)

• Almost commuting permutations are near commuting permutations
• Quasi-morphisms on Free Groups
• Surface groups are flexibly stable
• Abelian groups are polynomially stable
• On uniform Hilbert Schmidt stability of groups
• Poincaré series of Lie lattices and representation zeta functions of arithmetic groups
• Garland's method with Banach coefficients