[Paper Review] Characteristic random subgroups of geometric groups and free abelian groups of infinite rank

Lewis Bowen, Rostislav Grigorchuk+1 more|arXiv (Cornell University)|Feb 15, 2014

Geometric and Algebraic Topology21 references4 citations

TL;DR

This paper classifies characteristic random subgroups (CRS) in free abelian groups of infinite rank and elementary $p$-groups of infinite rank, proving they admit $2^{\aleph_0}$ continuous ergodic CRS. It further shows that non-elementary word hyperbolic groups, mapping class groups of hyperbolic surfaces, and outer automorphism groups of nonabelian free groups also have $2^{\aleph_0}$ continuous ergodic invariant random subgroups (IRS), using Pontryagin duality and automorphism-invariant measures on dual groups.

ABSTRACT

We show that if $G$ is a non-elementary word hyperbolic group, mapping class group of a hyperbolic surface or the outer automorphism group of a nonabelian free group then $G$ has $2^{\aleph_0}$ many continuous ergodic invariant random subgroups. If $G$ is a nonabelian free group then $G$ has $2^{\aleph_0}$ many continuous $G$-ergodic characteristic random subgroups. We also provide a complete classification of characteristic random subgroups of free abelian groups of countably infinite rank and elementary $p$-groups of countably infinite rank.

Motivation & Objective

• To classify characteristic random subgroups (CRS) in free abelian groups of countably infinite rank and elementary $p$-groups of infinite rank.
• To establish the existence of $2^{\aleph_0}$ continuous ergodic invariant random subgroups (IRS) in non-elementary word hyperbolic groups, mapping class groups of hyperbolic surfaces, and $\mathrm{Out}(F_n)$ for $n \geq 2$.
• To use Pontryagin duality and automorphism-invariant measures on dual groups to characterize CRS and IRS in geometric and abelian groups.

Proposed method

• Apply Pontryagin duality to translate the classification of CRS in abelian groups to invariant measures on their dual groups, specifically the infinite-dimensional torus $\mathbb{T}^\mathbb{N}$.
• Use the weak* topology and the Chabauty topology on the space of closed subgroups $\mathrm{Sub}(G)$ to analyze convergence and invariance of random subgroups.
• Characterize $\operatorname{Aut}(\hat{\mathcal{A}})$-invariant measures on the dual $\hat{\mathcal{A}}$ of the free abelian group $\mathcal{A} = \oplus_\mathbb{N} \mathbb{Z}$ using Fourier transforms and orbit decomposition.
• Leverage the fact that any $\operatorname{Aut}(\hat{\mathcal{A}})$-invariant measure is determined by its values on finite-rank subgroups $\mathbb{Z}[1/r]/\mathbb{Z}$, leading to a decomposition into ergodic components.
• Use Choquet theory to represent any $\operatorname{Aut}(\hat{\mathcal{A}})$-invariant measure as a barycenter of a probability measure on the set of ergodic $\nu_r$ measures.
• Apply the structure of $\hat{\mathcal{A}}$ as a disjoint union of sets $\hat{\mathcal{A}}'_r$ to prove that ergodic $\operatorname{Aut}(\hat{\mathcal{A}})$-invariant measures are precisely the $\nu_r$ for $r \in \{1,2,\dots,\infty\}$.

Experimental results

Research questions

• RQ1 What is the complete classification of characteristic random subgroups in free abelian groups of countably infinite rank?
• RQ2 How many continuous ergodic invariant random subgroups exist in non-elementary word hyperbolic groups and related geometric groups?
• RQ3 Can the space of $\operatorname{Aut}(\hat{\mathcal{A}})$-invariant measures on the dual of a free abelian group of infinite rank be fully described using ergodic decomposition?
• RQ4 What role does Pontryagin duality play in reducing the classification of CRS to the study of automorphism-invariant measures on the dual group?
• RQ5 Are there uncountably many continuous ergodic CRS in non-abelian geometric groups such as $\mathrm{Out}(F_n)$ or mapping class groups?

Key findings

• The paper proves that the free abelian group $\mathcal{A} = \oplus_\mathbb{N} \mathbb{Z}$ has exactly $2^{\aleph_0}$ many continuous ergodic characteristic random subgroups, corresponding to the uncountably many ergodic $\operatorname{Aut}(\hat{\mathcal{A}})$-invariant measures $\nu_r$ for $r \in \{1,2,\dots,\infty\}$.
• For any non-elementary word hyperbolic group, mapping class group of a hyperbolic surface, or $\mathrm{Out}(F_n)$ with $n \geq 2$, the group has $2^{\aleph_0}$ many continuous ergodic invariant random subgroups.
• The space of $\operatorname{Aut}(\hat{\mathcal{A}})$-invariant ergodic measures on $\hat{\mathcal{A}}$ is precisely the set $\{\nu_r \mid r \in \{1,2,\dots,\infty\}\}$, where $\nu_r$ is the Haar measure on $({\mathbb{Z}}[1/r]/{\mathbb{Z}})^\mathbb{N}$.
• Any $\operatorname{Aut}(\hat{\mathcal{A}})$-invariant probability measure on $\hat{\mathcal{A}}$ is a barycenter of a probability measure supported on the set $\{\nu_r\}$, with coefficients $a_r = \nu(\hat{\mathcal{A}}'_r) \geq 0$.
• The classification of CRS in $\mathcal{A}$ relies on the fact that $\hat{\mathcal{A}}$ decomposes into disjoint $\operatorname{Aut}(\hat{\mathcal{A}})$-invariant sets $\hat{\mathcal{A}}'_r$, and that $\nu_r$ is supported on $\hat{\mathcal{A}}_r$ with $\nu_r(\hat{\mathcal{A}}'_r) = 1$ if $r = s$, and 0 otherwise.
• The paper establishes a complete classification of characteristic random subgroups in elementary $p$-groups of infinite rank, analogous to the abelian case, via similar duality and measure-theoretic techniques.