Surface groups are flexibly stable

TL;DR

This paper demonstrates that surface groups exhibit flexible stability in permutations, using geometric methods involving hyperbolic surface covers, and introduces a quantitative LERF property for surface groups.

Contribution

It provides the first non-trivial example of a non-amenable flexibly stable group and develops a geometric approach to analyze stability via branched covers.

Findings

Surface groups are flexibly stable in permutations.

Established a quantitative LERF property for surface groups.

Introduced a geometric method for stability analysis.

Abstract

We show that surface groups are flexibly stable in permutations. This is the first non-trivial example of a non-amenable flexibly stable group. Our method is purely geometric and relies on an analysis of branched covers of hyperbolic surfaces. Along the way we establish a quantitative variant of the LERF property for surface groups which may be of independent interest.