Bonn GGT Seminar

Complete classification of the Dehn functions of Bestvina–Brady groups

Introduced by Bestvina and Brady in 1997, Bestvina–Brady groups form an important class of examples in geometric group theory and topology, known for exhibiting unusual finiteness properties. In this talk, I will focus on the Dehn functions of finitely presented Bestvina–Brady groups. The Dehn function of a group is a fundamental quasi-isometry invariant that, roughly speaking, provides a quantitative measure of the group’s finite presentability. After reviewing known results, I will present a classification of the Dehn functions of Bestvina–Brady groups. This talk is based on joint work with Yu-Chan Chang and Matteo Migliorini.

The Wiegold conjecture for the small Ree groups

Let $G$ be a finite simple group and $n \geq 3$. Is the action of $\mathrm{Aut}(F_n)$ on $\mathrm{Epi}(F_n,G)$ transitive? Research on this question dates back to 1977, and a positive answer is related to the so-called Wiegold conjecture. For a prime number $p$, a positive answer is known for the following groups: $\mathrm{PSL}(2,p)$, $\mathrm{Sz}(2^{2m-1})$ and $\mathrm{PSL}(2,2^m)$ for $m \geq 2$, $\mathrm{PSL}(2,3^p)$, $\mathrm{PSL}(2,p^r)$ for $r \geq 1$ when $n \geq 4$, and $\mathrm{Alt}(k)$ for $k \leq 11$ when $n = 3$. We showed that it can also be answered positively for the small Ree groups $\!^2 G_2 (3^{2e+1})$ for all $e \geq 1$ when $n \geq 5$. This was joint work with Mark Pengitore, Jeroen Schillewaert, and Hendrik Van Maldeghem. In this talk, we provide some insight into the conjecture, discuss some of the methods used to prove our result for the small Ree groups, and explain why it is not easy to resolve all cases.

Can you hear the shape of a Hyperbolic Marimba?

Hyperbolic surfaces have historically been played as drums, with little success. By work of Vignéras and Sunada, the frequencies at which a hyperbolic drum vibrates do not determine the surface up to isometry. In this talk, we are going to play a hyperbolic surface as a marimba: we assign notes to a finite collection of closed geodesics and, following a random geodesic on the surface, we play a note when the corresponding closed curve is hit. We investigate to what extent the melodies produced determine the hyperbolic surface. Joint work with Juan Souto.

Rigidity of embeddings between groups acting on spaces of dimension at most one

Using semi-conjugacies, I will talk about some embedding and non-embedding results for subgroups of $\mathrm{Homeo}_+(I)$ into Thompson's group $V$ via induced actions on the circle. This is joint work with James Hyde and Matt Zaremsky.