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.

Quasiisometric embeddings between cycle RAAGs

Given a graph, the associated right-angled Artin group (RAAG) has a generator for each vertex, and two generators commute when their vertices span an edge of the graph. These groups can be surprisingly complex even for fairly simple graphs. In this talk we consider the case where the graph is a cycle of length at least five. Huang showed that such RAAGs are quasiisometrically rigid, and Kim--Koberda understood when one is a subgroup of another. We'll describe when one can be quasiisometrically embedded in another, which turns out to be more often than you might expect. Based on joint work with Shaked Bader and Oussama Bensaid.

Incubulable hyperbolic 3-pseudomanifold groups

The fundamental group of a closed hyperbolic 3-manifold is known to act geometrically on a $\mathrm{CAT}(0)$ cube complex. We ask whether the same is true for the fundamental group of negatively curved 3-pseudomanifolds, i.e., 3-manifolds with isolated singularities and a locally $\mathrm{CAT(-1)}$ metric. While many 3-pseudomanifolds are cubulated, such as those arising from RACGs and strict hyperbolization, in this talk we give the first examples of closed 3-pseudomanifolds that are locally $\mathrm{CAT}(-1)$ but whose fundamental group cannot be cubulated. These examples are obtained from certain compact hyperbolic 3-manifolds with totally geodesic boundary by coning off the boundary components. This is joint work with Jason Manning.

A virtual fibering criterion for amalgamated free products

Let $G$ be a group acting on a tree. I will discuss necessary conditions for $G$ to have a finitely generated infinite normal subgroup of infinite index. When the edge stabilisers are virtually cyclic this naturally leads to considering (virtual) fibering of $G$. I will give an “if and only if” criterion for (virtual) fibering in the special case of amalgamated free products over virtually cyclic subgroups. The talk will be based on joint work with Jon Merladet.