Surface groups are flexibly stable

TL;DR

The paper proves that the fundamental group of any closed orientable surface $S$ with genus $g\ge 2$ is flexibly stable in permutations. It develops a geometric framework based on branched covers, $*$-covers, and a quantitative LERF-type result, combining Voronoi–Delaunay structures on singular hyperbolic planes with cut graphs to convert branched covers into unramified ones while controlling area and boundary length. A key contribution is a linear-bound capping-off theorem, yielding a explicit stable-to-near-stable transition: a near-action can be corrected to a genuine action after extending the permutation degree by at most a factor $(1+\varepsilon)$, with $\varepsilon$ scaling like $\delta\ln(1/\delta)$. This advances understanding of stability vs. soficity for hyperbolic groups and provides quantitative tools for approximating group actions by finite permutations via geometric constructions.

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.

Figures (3)

• Figure 1: The Voronoi cell $A_v$ based at the vertex $v$.
• Figure 2: The maps $r$ and $q$ are covers, the maps $f$ and $g$ are quotients, and the diagram is commutative. In this example $A = \{\alpha, \beta\}$ and $\bar{A}^\pm = \{\alpha^\pm, \beta^\pm\}$. The curve $\alpha$ satisfies $N_\alpha = M$ and the curve $\beta$ satisfies $N_\beta < M$.
• Figure 3: The simple closed curves $\gamma_0,\ldots,\gamma_{2g-1}$ are given by $\gamma_0 = e_0$, $\gamma_{2g-1} = e_{2g-1}$ and $\gamma_i = e_i \bar{e} '_i$ for every other $i$.

Theorems & Definitions (58)

• Definition
• Theorem 1.1
• Theorem 1.2
• Definition
• Definition
• Definition
• Definition
• Theorem 2.1
• Proposition 2.2
• proof : Proof of Proposition \ref{['prop:construction of *-cover from a homomorphism']}