Martín's webpage

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

Since September 2023 I am a PhD student in mathematics at Institut Camille Jordan in Lyon, France. My advisor is Nicolás Matte Bon.

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 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.

Preprints (click on the paper's title to display its abstract)

• 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 [Ergodic Theory Dynam. Systems, 2013] and A. Navas [Proceedings of the International Congress of Mathematicians, 2018].

  
  2025 preprint. arxiv, pdf.
  Here's a poster Eduardo made about this paper, presented at YGGT XIII on April 2025.


• 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 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.

  
  2024 preprint. arxiv, pdf.


• 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.

  
  2024 preprint. arxiv, pdf. Journal of Group Theory, to appear.