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Markov processes associated to fractal branch groups

Jorge Fariña-Asategui

TL;DR

The work studies measure-preserving dynamics from fractal branch profinite groups acting on regular rooted trees, using Bowen's $f$-invariant to identify Markov structure over free semigroups. For fractal finite-type groups, it proves $f(G)=\log|G_1|-r_D(G)$ and that $F(T,\alpha_s^n)=f(G)$ for all $n\ge D$, yielding a Markov process realized by $\alpha_s^D$. It develops a Hausdorff-dimension based classification: if $f(G)=f(H)$ and $\mathrm{hdim}_{\mathrm{Aut}~T}(G)=\mathrm{hdim}_{\mathrm{Aut}~T}(H)$, then the associated Markov processes are isomorphic. The results are illustrated via GGS-groups and universality phenomena, showing how symmetry of defining vectors affects invariants and enabling a countable family of isomorphism classes. Overall, the paper connects fractal group theory, dimension theory, and non-amenable dynamics to classify Markov processes over free semigroups and provide a practical criterion for measure-conjugacy.

Abstract

The author introduced recently a new natural construction which associates a measure-preserving dynamical system to any fractal profinite group. Here, we investigate these measure-preserving dynamical systems under the extra assumption on the groups to be branch. First, we compute their $f$-invariant, a measure-conjugacy invariant introduced by Bowen, and show that they are Markov processes over free semigroups in the sense of Bowen. Secondly, we show that fractal branch profinite groups with the same Hausdorff dimension and whose associated measure-preserving dynamical systems have the same $f$-invariant yield isomorphic Markov processes.

Markov processes associated to fractal branch groups

TL;DR

The work studies measure-preserving dynamics from fractal branch profinite groups acting on regular rooted trees, using Bowen's -invariant to identify Markov structure over free semigroups. For fractal finite-type groups, it proves and that for all , yielding a Markov process realized by . It develops a Hausdorff-dimension based classification: if and , then the associated Markov processes are isomorphic. The results are illustrated via GGS-groups and universality phenomena, showing how symmetry of defining vectors affects invariants and enabling a countable family of isomorphism classes. Overall, the paper connects fractal group theory, dimension theory, and non-amenable dynamics to classify Markov processes over free semigroups and provide a practical criterion for measure-conjugacy.

Abstract

The author introduced recently a new natural construction which associates a measure-preserving dynamical system to any fractal profinite group. Here, we investigate these measure-preserving dynamical systems under the extra assumption on the groups to be branch. First, we compute their -invariant, a measure-conjugacy invariant introduced by Bowen, and show that they are Markov processes over free semigroups in the sense of Bowen. Secondly, we show that fractal branch profinite groups with the same Hausdorff dimension and whose associated measure-preserving dynamical systems have the same -invariant yield isomorphic Markov processes.

Paper Structure

This paper contains 15 sections, 9 theorems, 41 equations.

Table of Contents

  1. introduction
  2. Fractal groups and measure-preserving dynamical systems
  3. Groups acting on regular rooted trees
  4. Measure-preserving dynamical systems
  5. The $f$-invariant
  6. Markov processes
  7. Fractal groups of finite type
  8. Groups of finite type
  9. The $f$-invariant of fractal groups of finite type
  10. The Hausdorff dimension of self-similar groups
  11. A criterion for measure-conjugacy
  12. Applications and examples
  13. Universality of the groups $G_\mathcal{S}$
  14. Non-fractal groups of finite type
  15. An example: GGS-groups

Key Result

Theorem A

Let $G\le \mathrm{Aut}~T$ be a fractal branch profinite group. Then, there exists some $D\ge 1$ such that the $f$-invariant of the measure-preserving dynamical system $(G,\mu_G,T,\mathcal{T})$ is given by In particular, the process $(G,\mu_G,T,\mathcal{T},\alpha_s^D)$, where $\alpha_s$ denotes the standard partition of $(G,\mu_G)$, is Markov.

Theorems & Definitions (16)

  • Theorem A
  • Theorem B
  • Definition 2.1: Isomorphism of processes
  • Theorem 2.2: see JorgeCyclicity
  • Definition 2.3: Past of a vertex
  • Definition 2.4: Markov process BowenMarkov
  • Theorem 2.5: see BowenMarkov
  • Theorem 3.1: see SunicHausdorff and GrigorchukFinite
  • Lemma 3.2
  • proof
  • ...and 6 more