Continuum of allosteric actions for non-amenable surface groups

TL;DR

This paper constructs a continuum of ergodic, minimal, profinite actions for surface groups that are topologically free but not essentially free, revealing a rich variety of allosteric behaviors.

Contribution

It introduces the concept of allostery in the context of surface group actions and constructs a continuum of pairwise distinct IRSs with this property.

Findings

Existence of a continuum of allosteric actions for surface groups.

These actions are ergodic, minimal, and topologically free.

The IRSs obtained are pairwise distinct.

Abstract

Let $\Sigma$ be a closed surface other than the sphere, the torus, the projective plane or the Klein bottle. We construct a continuum of p.m.p. ergodic minimal profinite actions for the fundamental group of $\Sigma$, that are topologically free but not essentially free, a property that we call allostery. Moreover, the IRS's we obtain are pairwise distincts.