Since their introduction by Gromov in the 80s, a wealth of tools have been developed to study hyperbolic groups. Thus, when studying a class of groups, a characterisation of those that are hyperbolic can be very useful. In this talk, we will turn to the class of one-relator groups. In previous work, we showed that a one-relator group not containing any Baumslag--Solitar subgroups is hyperbolic, provided it has a Magnus hierarchy in which no one-relator group with a so called `exceptional intersection' appears. I will define one-relator groups with exceptional intersection, discuss the aforementioned result and will then provide a characterisation of the hyperbolic one-relator groups with exceptional intersection. Finally, I will then discuss how this characterisation can be used to establish properties for all one-relator groups.

Various graphs associated to surfaces have proved to be important tools for studying the large scale geometry of mapping class groups of surfaces, among other applications. A seminal paper of Masur and Minsky proved that perhaps the most well known example, the curve graph, is Gromov hyperbolic. However, this is not the case for every naturally defined graph associated to a surface. We will present joint work with Jacob Russell classifying a wide family of graphs associated to surfaces according to whether the graph is Gromov hyperbolic, relatively hyperbolic or not relatively hyperbolic.
 

I will discuss some new invariants of 2-complexes. They are inspired by recent developments in the theory of one-relator groups, but also have the potential to unify the theories of many well-studied families including small-cancellation presentation complexes, CAT(0) 2-complexes and 3-manifold spines, in addition to the motivating examples of one-relator presentation complexes. The fundamental result is that these invariants are the extrema of explicit linear-programming problems, and in particular are rational, computable and realised. The definitions suggest a conjectural “map” of 2-complexes, which I will attempt to describe.
 

Consider a random group $\Gamma$ with $k$ generators and $r$ random relators of large length $N$. We study the geometry of the character variety of $\Gamma$ with values in $\SL(2,\C)$ or more generally any semisimple Lie group $G$. This is the moduli space of group homomorphisms from $\Gamma$ to $G$ up to conjugation. We are in particular able to determine its dimension, number of components and Galois group, with an excellent control on the probability of exceptions. The proofs use effective Chebotarev type theorems as well as new spectral gap bounds  for Cayley graphs of finite simple groups. They are also conditional on GRH. Joint work with Peter Varju and Oren Becker.