$Out(F_n)$-invariant probability measures on the space of $n$-generated marked groups

TL;DR

This paper demonstrates the existence of numerous non-atomic, invariant probability measures on the space of n-generated marked groups, revealing complex symmetry properties and implications for model theory.

Contribution

It establishes the existence of a vast family of invariant measures under Out(F_n) and explores their implications, highlighting the role of acylindrical hyperbolicity.

Findings

Existence of 2^{aleph_0} non-atomic, invariant, mixing probability measures on _n.

Some closed subsets of _n admit no invariant probability measures.

Connections between invariant measures and model theoretic properties.

Abstract

Let $\mathcal G_n$ denote the space of $n$-generated marked groups. We prove that, for every $n\ge 2$, there exist $2^{\aleph_0}$ non-atomic, $Out(F_n)$-invariant, mixing probability measures on $\mathcal G_n$. On the other hand, there are non-empty closed subsets of $\mathcal G_n$ that admit no $Out(F_n)$-invariant probability measure. Acylindrical hyperbolicity of the group $Aut(F_n)$ plays a crucial role in the proof of both results. We also discuss model theoretic implications of the existence of $Out(F_n)$-invariant, ergodic probability measures on $\mathcal G_n$.