L6

The space of measured laminations on a hyperbolic surface is a generalisation of the set of weighted multi curves. The action of the mapping class group on this space is an important tool in the study of the geometry of the surface. 
For orientable surfaces, orbit closures are now well-understood and were classified by Lindenstrauss and Mirzakhani. In particular, it is one of the pillars of Mirzakhani’s curve counting theorems. 
For non-orientable surfaces, the behaviour of this action is very different and the classification fails. In this talk I will review some of these differences and describe mapping class group orbit closures of (projective) measured laminations for non-orientable surfaces. This is joint work with Erlandsson, Gendulphe and Souto.

If a group quasiisometrically embeds into a finite product of infinite valence trees then a number of things are implied; for example, the group will have finite Assouad-Nagata dimension and finite asymptotic dimension. An even stronger statement is that the group quasiisometrically embeds into a finite product of uniformly bounded valence trees. The research on which groups quasiisometrically embed into finite products of uniformly bounded valence trees is limited, however a notable result of Buyalo, Dranishnikov and Schroeder from 2007 proves that all hyperbolic groups do admit these quasiisometric embeddings. In a recently released preprint, I extend their result to cover groups which are relatively hyperbolic with respect to virtually abelian peripheral subgroups. 

This talk will focus on the ideas at the core of Buyalo, Dranishnikov and Schroeder’s result and the extension that I proved, and in particular I will attempt to provide a general framework for upgrading quasiisometric embeddings into infinite valence trees so that they are now quasiisometric embeddings into uniformly bounded valence trees. The central concept is called a diary which I will define. 

Measure equivalence was introduced by Gromov as a measure-theoretic analogue to quasi-isometry between finitely generated groups. In this talk I will present measure equivalence classification results for right-angled Artin groups, and more generally graph products. This is based on joint works with Jingyin Huang and with Amandine Escalier. 

We will give a relaxed introduction to some of the most classical dynamical systems - Anosov flows. These flows were highly influential in the development of ideas which the audience might be more familiar with. For example, Anosov flows give rise to exponential group growth and taut foliations, both of which we will discuss. Finally, we will talk about some recent work obstructing Anosov flows and their combinatorial analogs - veering triangulations

Property (T) was introduced by Kazhdan in the sixties to show that lattices in higher rank semisimple Lie groups are finitely generated. We will discuss some classical examples of groups that satisfy this property, with a particular focus on SL(3, R).

Graph braid groups are similar to braid groups, except that they are defined as ‘braids’ on a graph, rather than the real plane. We can think of graph braid groups in terms of the discrete configuration space of a graph, which is a CW-complex. One can compute a presentation of a graph braid group using Morse theory. In this talk I will give a few examples on how to compute these presentations in terms of generating circuits of the graph. I will then go through a detailed example of a graph that gives a one-ended braid group.

For a hyperbolic knot there are two types of invariants, the hyperbolic invariants coming from the geometric structure and the classical invariants coming from the topology or combinatorics. It has been observed in many different cases that these seemingly different types of invariants are in fact related. I will give examples of these relationships and discuss in particular a link by Stoimenow between the determinant and volume.