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

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, $\mathrm{Out}(F_n)$-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 $\mathrm{Out}(F_n)$-invariant probability measure. Acylindrical hyperbolicity of the group $\mathrm{Aut}(F_n)$ plays a crucial role in the proof of both results. We also discuss model-theoretic implications of the existence of $\mathrm{Out}(F_n)$-invariant, ergodic probability measures on ${\mathcal{G}}_n$.