Local permutation stability

TL;DR

This paper introduces a new, weaker notion of local stability in permutations for finitely generated groups, linking it to LEF groups and providing criteria for amenable groups, with applications to topological full groups.

Contribution

It defines local permutation stability, establishes its relation to LEF groups, and characterizes it for amenable groups using invariant random subgroups, with applications to topological full groups.

Findings

Locally stable sofic groups are LEF.

A necessary and sufficient condition for amenable groups to be locally permutation stable.

Derived subgroups of topological full groups are locally stable, yielding many non-stable examples.

Abstract

We introduce a notion of "local stability in permutations" for finitely generated groups. If a group is sofic and locally stable in our sense, then it is also locally embeddable into finite groups (LEF). Our notion is weaker than the "permutation stability" introduced by Glebsky-Rivera and Arzhantseva-Paunescu, which allows one to upgrade soficity to residual finiteness. We prove a necessary and sufficient condition for an amenable group to be locally permutation stable, in terms of invariant random subgroups (IRSs), inspired by a similar criterion for permutation stability due to Becker, Lubotzky and Thom. We apply our criterion to prove that derived subgroups of topological full groups of Cantor minimal subshifts are locally stable, using Zheng's classification of IRSs for these groups. This last result provides continuum-many groups which are locally stable, but not stable.