Markov processes associated to fractal branch groups

2025/05/27 by Fariña-Asategui, Jorge · 1 citation
#22F50 #60J05. Secondary: 37A05 #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR) #Primary: 20E08

paper · doi:10.48550/arxiv.2505.21134

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.

Citations

• On the fixed-point proportion of self-similar groups
• On the Basilica Operation
• GGS-groups: order of congruence quotients and Hausdorff dimension
• A measure-conjugacy invariant for free group actions
• Hausdorff dimension in a family of self-similar groups
• Self-similar groups and topological dimension

Cited by

• Random subgroups of branch groups

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