Tag - Hyperbolic groups

Laura Ciobanu: Group equations, constraints and decidability

8th March, 2024

In this talk I will discuss group equations with non-rational constraints, a topic inspired by the long line of work on word equations with length constraints. Deciding algorithmically whether a word equation has solutions satisfying linear length constraints is a major open question, with deep theoretical and practical implications. I will introduce equations in groups and several kinds of constraints, and show that equations with length, abelian or context-free constraints are decidable in virtually abelian groups (joint with Alex Evetts and Alex Levine). This contrasts the fact that solving equations with abelian constraints is undecidable for non-abelian right-angled Artin groups and hyperbolic groups with ‘large’ abelianisation (joint work with Albert Garreta).

Giulio Tiozzo: Introduction to Random Walks on Groups

12th September, 2023

This is a 21-lecture course, with each lecture being either one or two hours, given by Giulio Tiozzo. It gives an introduction to random walks on groups. This class will focus on properties of group actions from a probabilistic point of view, investigating the relations between the dynamics, measure theory and geometry of groups.

We will start with a brief introduction to ergodic theory, discussing measurable transformations and the basic ergodic theorems. Then we will approach random walks on matrix groups and lattices in Lie groups, following the work of Furstenberg. Topics of discussion will be: positivity of drift and Lyapunov exponents. Stationary measures. Geodesic tracking. Entropy of random walks. The Poisson-Furstenberg boundary. Applications to rigidity. We will then turn to a similar study of group actions which do not arise from homogeneous spaces, but which display some features of negatively curved spaces: for instance, hyperbolic groups (in the sense of Gromov) and groups acting on hyperbolic spaces. This will lead us to applications to geometric topology: in particular, to the study of mapping class groups and Out(FN).

Prerequisites: An introduction to measure theory and/or probability, basic topology and basic group theory. No previous knowledge of geometric group theory or Teichmüller theory is needed.

Jeroen Schillewaert: Constructing highly regular expanders from hyperbolic Coxeter groups

17th November, 2022

Given a string Coxeter system (W,S), we construct highly regular quotients of the 1-skeleton of its universal polytope P, which form an infinite family of expander graphs when (W,S) is indefinite and P has finite vertex links. The regularity of the graphs in this family depends on the Coxeter diagram of (W,S). The expansion stems from superapproximation applied to (W,S). This construction is also extended to cover Wythoffian polytopes. As a direct application, we obtain several notable families of expander graphs with high levels of regularity, answering in particular a question posed by Chapman, Linial and Peled positively.

This talk is based on joint work with Marston Conder, Alexander Lubotzky and Francois Thilmany.

This video was produced by the Sydney Mathematical Research Institute, as part of their SMRI seminar series.

Kane Townsend: Hyperbolic groups with k-geodetic Cayley graphs

7th October, 2022

A locally-finite simple connected graph is said to be k-geodetic for some k ≥ 1, if there are at most k distinct geodesics between any two vertices of the graph. We investigate the properties of hyperbolic groups with k-geodetic Cayley graphs. To begin, we show that k-geodetic graphs cannot have a "ladder-like" geodesic structure with unbounded length. Using this bound, we generalize a well-known result of Papasoglu that states hyperbolic groups with 1-geodetic Cayley graphs are virtually free. We then investigate which elements of the hyperbolic group with k-geodetic Cayley graph commute with a given infinite order element.

Marco Linton: Hyperbolicity of Certain One-Relator Groups

22nd April, 2022

The primitivity rank of an element w of a free group F is defined as the minimal rank of a subgroup containing w as an imprimitive element. Recent work of Louder and Wilton has shown that there is a strong connection between this quantity and the subgroup structure of the one-relator group F/≪w≫. In particular, they show that one-relator groups whose defining relation has primitivity rank at least 3 cannot contain Baumslag-Solitar subgroups, leading them to conjecture that such groups are hyperbolic. In this talk, I will show how to confirm and strengthen this conjecture, providing some applications.

Ilya Kapovich: Primitivity rank for random elements in free groups

25th March, 2022

In 2014 Doron Puder introduced the notion of primitivity rank π(g) for a non-trivial element g in a free group Fr of rank r.

Namely, π(g) is defined as the smallest rank of a subgroup H of Fr containing g as a non-primitive element, or as ∞ if no such H exists. The set of all subgroups H of Fr as above is denoted Crit(g). It turns out that the primitivity index of an element w ∈ Fr is closely related to the questions about word-hyperbolicity and subgroup properties of the one-relator group < Fr | w=1 >.

We prove that if r≥2 and F2=F(x1, ..., xr) is the free group of rank r, then, as n→∞, for a "random" element wn ∈ Fr of length n with probability tending to 1 one has π(w)=r and Crit(w)={Fr}. We discuss applications of this result to "word measures" on finite symmetric groups SN, defined by such wn.

Sarah Rees: The compressed word problem in relatively hyperbolic groups

3rd December, 2021

I'll discuss recent work with Derek Holt that proves that the compressed word problem in groups that are hyperbolic relative to free abelian subgroups can be solved in polynomial time. This result extends results of Lohrey, and of Holt, Lohrey and Schleimer, for free groups and for word hyperbolic groups, and our proof imitates the proofs of those results. I'll define all the terms used in the title, explain background that motivates the result, and outline the methods used in the proof.

Suraj Krishna: The mapping torus of a torsion-free hyperbolic group is relatively hyperbolic

15th November, 2021

Let G be the fundamental group of a closed orientable surface of genus at least 2, and α an automorphism of G. In a celebrated result, Thurston showed that the mapping torus G⋊αℤ is hyperbolic if and only if no power of α preserves a non-trivial conjugacy class. In this talk, I will describe joint work with François Dahmani, where we show that if G is torsion-free hyperbolic, then G⋊αℤ is relatively hyperbolic with optimal parabolic subgroups.

John Mackay: Conformal dimension and decompositions of hyperbolic groups

22nd October, 2021

The boundary of a Gromov hyperbolic group carries a canonical family of metrics which determine the quasi-isometry type of the group. Pansu's conformal dimension of the boundary gives a natural and important quasi-isometric invariant. I will discuss how this invariant behaves when the group splits over two-ended subgroups (i.e. when the boundary has local cut points), and applications to the question of Bonk and Kleiner asking for a characterization of when this dimension equals one.