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$Out(F_n)$-invariant probability measures on the space of $n$-generated marked groups (2207.03659v4)

Published 8 Jul 2022 in math.GR and math.DS

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 $\mathcal G_n$.

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