Artin groups, mapping class groups and Out(Fn): from geometry to operator algebras via measure equivalence

Introduction

Negative curvature features of Artin groups, surface mapping class groups, and outer automorphisms of free groups have been the subject of intense study in geometric group theory. I propose to confront this viewpoint with recent developments in measured group theory and operator algebras. The focus is on obtaining structural and rigidity theorems in measure equivalence, and for von Neumann algebras associated with these groups and their ergodic actions.

First Goal

A first goal is to pursue the classification of right-angled Artin groups up to measure equivalence. This includes:

1. Describing the class of groups that are measure equivalent to a given one.
2. Tackling the question of their rigidity for integrable measure equivalence.

Beyond the right-angled case, I aim for new superrigidity theorems, both for quasi-isometry and measure equivalence, for many Artin groups (of hyperbolic type, of FC type). This will require describing lattices in the automorphism group of their Cayley complex.

Second Goal

A second goal is to prove the proper proximality of Out(Fn) in the sense of Boutonnet, Ioana, and Peterson. This would yield strong rigidity results for von Neumann algebras associated with its compact actions.

This will require developing an analogue for Out(Fn) to the Masur-Minsky theory for mapping class groups. These new tools, as a by-product, are likely to be helpful in tackling some of the most challenging questions on Out(Fn), like the Farrell-Jones conjecture.

Biggest Challenge

The biggest challenge will be to investigate the group von Neumann algebra L(G), when G is an Artin group, a mapping class group, or Out(Fn). I will start with the W*-classification of right-angled Artin groups with finite outer automorphism groups.

Ultimately, I aim at proving structural properties of L(G), like primeness, when G is as above and, as a long-term goal, establish rigidity phenomena.