Tag - Geometric group theory

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.

Claudio Bravo: Quotients of the Bruhat-Tits tree function field analogues of the Hecke congruence subgroups

1st July, 2022

Let C be a smooth, projective, and geometrically connected curve defined over a finite field F. For each closed point P∞ of C, let R be the ring of functions that are regular outside P∞, and let K be the completion path P∞ of the function field of C. In order to study groups of the form GL2(R), Serre describes the quotient graph GL2(R)∖T, where T is the Bruhat-Tits tree defined from SL2(K). In particular, Serre shows that GL2(R)∖T is the union of a finite graph and a finite number of ray shaped subgraphs, which are called cusps. It is not hard to see that finite index subgroups inherit this property. In this exposition we describe the quotient graph H∖T defined from the action on T of the group H consisting of matrices that are upper triangular modulo I, where I is an ideal of R. More specifically, we give an explicit formula for the cusp number H∖T. Then By, using Bass-Serre theory, we describe the combinatorial structure of H. These groups play, in the function field context, the same role as the Hecke Congruence subgroups of SL2(ℤ). Moreover, not that the groups studied by Serre correspond to the case where the ideal I coincides with the ring R.

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.

Jingyin Huang: The Helly geometry of some Garside and Artin groups

15th November, 2021

Artin groups emerged from the study of braid groups and complex hyperplane arrangements. Artin groups have very simple presentation, yet rather mysterious geometry with many basic questions widely open. I will present a way of understanding certain Artin groups and Garside groups by building geometric models on which they act. These geometric models are non-positively curved in an appropriate sense, and such curvature structure yields several new results on the algorithmic, topological and geometric aspects of these groups. No previous knowledge on Artin groups or Garside groups is required.

Alessandro Sisto: A simple hierarchical hyperbolicity criterion and extra-large Artin groups

5th November, 2021

A hierarchically hyperbolic structure is some kind of coordinate system on a given metric spaces where the coordinates take values in hyperbolic spaces, and it gives a good understanding of the coarse geometry of the space. I will give a brief introduction to this notion and its consequences, discuss a simple criterion to show that a space or group is hierarchically hyperbolic, and illustrate an application of this criterion to the case of extra-large type Artin groups.

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.