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

TL;DR

The paper addresses the existence and structure of $\operatorname{Out}(F_n)$-invariant probability measures on the space $\mathcal{G}_n$ of $n$-generated marked groups. It develops a general framework via acylindrically hyperbolic groups and wreath-like products to embed $2^{\operatorname{Out}(F_n)}$ into $\mathcal{G}_n$, thereby transferring $2^{\aleph_0}$ distinct invariant mixing measures from $2^{\operatorname{Out}(F_n)}$ to $\mathcal{G}_n$; it also shows that some closed subsystems admit no invariant measure. The work further connects these dynamical phenomena to model-theoretic consequences: ergodic invariant measures yield uncountably many pairwise non-isomorphic finitely generated groups that are elementarily equivalent, with additional results about the supports of such measures (e.g., restrictions on hyperbolic groups in supports). Together, these results illuminate the richness of the Out$(F_n)$-action on marked groups and highlight significant links between dynamics, group theory, and model theory.

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 $\mathcal G_n$.

Theorems & Definitions (31)

• Remark 1.2
• Theorem 1.4
• Corollary 1.5
• Corollary 1.6
• Lemma 2.1: KT and FKSV
• Definition 2.2
• Lemma 2.3: DGO
• Lemma 2.4
• proof
• Lemma 2.5: DGO