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Continuum of allosteric actions for non-amenable surface groups

Matthieu Joseph

Abstract

Let $Σ$ 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 $Σ$, that are topologically free but not essentially free, a property that we call allostery. Moreover, the IRS's we obtain are pairwise distincts.

Continuum of allosteric actions for non-amenable surface groups

Abstract

Let 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 , that are topologically free but not essentially free, a property that we call allostery. Moreover, the IRS's we obtain are pairwise distincts.

Paper Structure

This paper contains 13 sections, 17 theorems, 23 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments.
  3. Preliminaries
  4. Topological dynamic and URS/IRS
  5. Profinite actions and their URS/IRS
  6. Allostery and commensurability
  7. Finite index subgroups of surface groups
  8. Residual properties of surface groups
  9. Special kind of finite index subgroups in surface groups
  10. Step 1: Cyclic covering.
  11. Step 2: Erasing the right amount of generators.
  12. Step 3: The group $\Lambda_d/N$ is a residually finite $p$-group.
  13. Proof of the main theorem

Key Result

Theorem 1.2

Any non-amenable surface group admits a continuum of profinite allosteric actions that are pairwise topologically and measurably non-isomorphic.

Figures (3)

  • Figure 1: An illustration of the morphism that erases generators.
  • Figure 2: Illustrations of the proof of Theorem \ref{['thm.sousgroupe']}. The above line illustrates the coverings corresponding to the inclusions $\Lambda\leq\Lambda_d\leq\Gamma$. The bottom line illustrates the covering corresponding to the inclusion $N'\leq \Lambda_d/N$.
  • Figure 3: A branched surface

Theorems & Definitions (39)

  • Definition 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • Lemma 2.5
  • Lemma 2.6
  • ...and 29 more