arXiv subjects

Martín Gilabert Vio

Publications and source records attributed to Martín Gilabert Vio.

5 recordsLinked to original sources

arXiv2026

Groups with classifiable actions on the line

We motivate and study the class $\mathcal{C}$ of countable groups $G$ such that the conjugacy relation between minimal actions of $G$ on $\mathbb{R}$ by orientation-preserving homeomorphisms is smooth -- that is, admits a Borel transversal. No example of amenable group outside of $\mathcal{C}$ is known. We show a number of stability properties of $\mathcal{C}$ under group-theoretic operations and that $\mathcal{C}$ contains all finitely generated groups of piecewise affine homeomorphisms of the interval. We exhibit a finitely generated group $G$ that is not in $\mathcal{C}$, such that $G$ is amenable if and only if Thompson's group $F$ is amenable. We also prove that the semiconjugacy relation among cocompact actions of a countable group $G$ is smooth if and only if $G \in \mathcal{C}$, and that it is essentially countable even when $G$ is not finitely generated. In the Appendix, we show that there is no good analogue of the space of harmonic actions for a countable non-finitely generated group.

math.GR↗

arXiv2026

Stationary boundaries on the space of amenable subgroups and C*-simplicity

We give a sufficient condition for a countable group $G$ to possess a probability measure $μ$ that admits a non-trivial $μ$-boundary modeled in the space $\mathrm{Sub}_{\mathrm{am}}(G)$ of amenable subgroups of $G$. In particular, for such $μ$ the space $\mathrm{Sub}_{\mathrm{am}}(G)$ is not uniquely $μ$-stationary. This contrasts with a theorem of Hartman-Kalantar, which states that a countable group $G$ is C*-simple if and only if there exists $μ\in \mathrm{Prob}(G)$ such that $\mathrm{Sub}_{\mathrm{am}}(G)$ is uniquely $μ$-stationary. Our criterion applies to (permutational) wreath products, which include groups that are C*-simple, and to Thompson's group $F$, whose C*-simplicity is equivalent to its non-amenability and therefore remains an open problem. We also show that any non-trivial $μ$-boundary modeled on $\mathrm{Sub}_{\mathrm{am}}(G)$ is supported on amenable normalish subgroups, in the sense of Breuillard-Kalantar-Kennedy-Ozawa. As a consequence, we conclude that a countable group with no finite normal subgroups and no amenable normalish subgroups acts essentially freely on all its Poisson boundaries.

math.GR↗

arXiv2025

Dynamical Tits alternative for groups of almost automorphisms of trees

We prove a dynamical variant of the Tits alternative for the group of almost automorphisms of a locally finite tree $\mathcal{T}$: a group of almost automorphisms of $\mathcal{T}$ either contains a nonabelian free group playing ping-pong on the boundary $\partial \mathcal{T}$, or the action of the group on $\partial \mathcal{T}$ preserves a probability measure. This generalises to all groups of tree almost automorphisms a result of S. Hurtado and E. Militon for Thompson's group $V$, with a hopefully simpler proof.

math.GR↗

arXiv2025

The Poisson boundary of Thompson's group $T$ is not the circle

Let $μ$ be a nondegenerate probability measure with finite entropy on a countable group $G \leq \mathrm{Homeo}_+(S^1)$ of orientation-preserving homeomorphisms of the circle acting proximally, minimally and topologically nonfreely on $S^1$. We prove that the circle $S^1$ endowed with its unique $μ$-stationary probability measure is not the Poisson boundary of $(G,μ)$. When $G$ is Thompson's group $T$ and $μ$ is finitely supported, this answers a question posed by B. Deroin [Ergodic Theory Dynam. Systems, 2013] and A. Navas [Proceedings of the International Congress of Mathematicians, 2018].

math.DS↗

arXiv2025

Probabilistic Tits alternative for circle diffeomorphisms

Let $μ_1, μ_2$ be probability measures on $\mathrm{Diff}^1_+(S^1)$ satisfying a suitable moment condition and such that their supports genererate discrete groups acting proximally on $S^1$. Let $(f^n_ω)_{n \in \mathbb{N}}, (f^n_{ω'})_{n \in \mathbb{N}}$ be two independent realizations of the random walk driven by $μ_1, μ_2$ respectively. We show that almost surely there is an $N \in \mathbb{N}$ such that for all $n \geq N$ the elements $f^n_ω, f^n_{ω'}$ generate a nonabelian free group. The proof is inspired by the strategy by R. Aoun for linear groups and uses work of A. Gorodetski, V. Kleptsyn and G. Monakov, and of P. Barrientos and D. Malicet. A weaker (and easier) statement holds for measures supported on $\mathrm{Homeo}_+(S^1)$ with no moment conditions.

math.GR↗