Martín's webpage

Hello, my name is Martín Gilabert Vio and this is my academic webpage.

From September 2023 to June 2026 I was a PhD student in mathematics at Institut Camille Jordan in Lyon, France, under the supervision of Nicolás Matte Bon. From September 2026 onwards I will be a postdoc at Institut de recherche mathématique de Rennes in Rennes, France.

I am interested in the intersection of group theory, dynamics and probability. I like thinking about Thompson groups, groups of homeomorphisms of 1-manifolds, random walks on infinite groups, Chabauty dynamics and more.

My professional email is gilabert at math dot univ-lyon1 dot fr. My office is bureau 131, bâtiment Braconnier, Institut Camille Jordan, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne, France.

Publications and preprints (click on the titles to display more info)

• Groups with classifiable actions on the line (with Joaquín Brum and Nicolás Matte Bon)

  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. 46 pp.

  Slides.

  
  2026 preprint. doi, pdf.


• Stationary boundaries on the space of amenable subgroups and C*-simplicity (with Anna Cascioli and Eduardo Silva)

  We give a sufficient condition for a countable group \(G\) to possess a probability measure \(\mu\) that admits a non-trivial \(\mu\)-boundary modeled in the space \(\mathrm{Sub}_\mathrm{am}(G)\) of amenable subgroups of \(G\). In particular, for such \(\mu\) the space \(\mathrm{Sub}_\mathrm{am}(G)\) is not uniquely 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 \(\mu \in \mathrm{Prob}(G)\) 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 \(\mu\)-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. 26 pp.

  
  2026 preprint. doi, pdf.


• The Poisson boundary of Thompson's group \(T\) is not the circle (with Cosmas Kravaris and Eduardo Silva)

  Let \(\mu\) be a nondegenerate probability measure with finite entropy on a countable group \(G\) 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 \(\mu\)-stationary probability measure is not the Poisson boundary of \( (G, \mu) \). When \(G\) is Thompson's group \(T\) and \(\mu\) is finitely supported, this answers a question posed by B. Deroin and by A. Navas. 34 pp.

  Here's a poster Eduardo made about this paper. Slides.

  
  2025 preprint. doi, pdf.


• Probabilistic Tits alternative for circle diffeomorphisms

  Let \(\mu_1,\mu_2\) be probability measures on \( \mathrm{Diff}^1_+(S^1) \) satisfying a suitable moment condition and such that their supports genererate countable groups acting proximally on \(S^1\). Let \( (f^n_\omega)_{n \in \mathbb{N}},\, (f^n_{\omega'})_{n \in \mathbb{N}} \) be two independent realizations of the random walk driven by \( \mu_1, \mu_2 \) respectively. We show that almost surely there is an \( N \in \mathbb{N} \) such that for all \( n \geq N \) the elements \( f^n_\omega, \, f^n_{\omega'} \) generate a nonabelian free group. The proof is inspired by the strategy by R. Aoun for linear groups. A weaker (and easier) statement holds for measures supported on \( \mathrm{Homeo}_+(S^1) \) with no moment conditions. 20 pp.

  After this preprint appeared, Inhyeok Choi generalized its results by adapting Gouëzel's pivoting technique to this context, see here.

  
  2024 preprint. doi, pdf. Commentarii Mathematici Helvetici, to appear.


• 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. 12 pp.

  
  2024 preprint. doi, pdf. Journal of Group Theory, Volume 29, Issue 1 (2026).

My preprints can be found on the arxiv too.

Math links: arxiv, HAL, zbMATH, MR Lookup, Detexify, VimTeX.

Cool stuff: Kiwis by Beat!, bitter films, Amanita Design.