Schedule

We discuss the ideas behind the proof of the following statement. For any hyperbolic group G and natural numbers r₁ ,r₂, there exists r = r(r₁,r₂,G) such that rank(H₁ ∩ H₂) ≤ r for any two locally quasiconvex subgroups H₁,H₂ < G of ranks r₁,r₂. This is based on joint work with Nir Lazarovich.

Hierarchically hyperbolic groups form a vast family of groups satisfying a coarse but highly organised form of non-positive curvature. Notable examples include hyperbolic groups, mapping class groups, compact special groups and many Artin groups. In this talk, I will discuss recent joint work with Mark Hagen in which we investigated the behaviour of their virtually abelian subgroups, proving in particular a coarse analogue of the CAT(0) flat torus theorem and flat closing problem for this class of groups.

Joint work with Edmund Heng. I will explain how to define linear representations of Coxeter groups over fusion rings and how such representations provide a geometric interpretation for certain homomorphisms, known as LCM homomorphisms, between Artin groups.

We exhibit a compact 4-dimensional metric space X that is locally CAT(0), and whose fundamental group is not hyperbolic and does not contain Z x Z. These properties are deduced by classifying the 2-dimensional flats in the universal cover of X. 

We will discuss our recent solution to the Twist Conjecture, which, combined with earlier work of Mühlherr with Howlett and Marquis, resolves the isomorphism problem for Coxeter groups. Other applications include finite generation of automorphism groups of Coxeter groups, as well as a substantial reduction for the isomorphism problem for Artin groups.

(Loint with Sam Hughes and Wolfgang Lueck) We give examples to show that theorems of S. Fisher and D. Kielak relating the vanishing of L² cohomology to fibering over the circle for RFRS groups cannot be extended to some larger classes of groups. Time permitting I may also discuss an L² torsion invariant (introduced by Hughes–Lueck alone) that may prove useful.

Property (Tₙ) is a group property introduced by U. Bader and R. Sauer, defined by the vanishing of the cohomology groups of G with L²-coefficients up to degree n. They showed that simple Lie groups have property (Tₙ) for n = rank−1. We prove a similar result for Lᵖ-coefficients, confirming a conjecture of Gromov in the special case of Lᵖ(G). The case p=1 is of particular interest. We relate the vanishing of L¹-cohomology to a fixed point theorem and confirm a conjecture of Farb: Farb conjectured a fixed point property for actions of lattices in such groups on CAT(0) cell complexes of dimension lower than the rank. In this talk, I will give a short introduction to property (T) and discuss ideas from the proofs of both results. This talk is based on joint work with Saar Bader, Uri Bader, and Roman Sauer.