Series Title: RAAG-Out: a savoury geometric group theory sampler
Series Blurb:
Coxeter and Artin groups are ubiquitous in mathematics, with many connections to algebra, geometry, topology, and representation theory. They have beautiful presentations which you can define via a labelled graph. While Coxeter groups are well behaved, Artin groups are more mysterious in nature but include some of our best friends: braid groups, free groups and (a best buddy to anyone who works with cube complexes) Right Angled Artin Groups (RAAGs). This mini course will explore Coxeter and Artin groups in general, the geometric tool we use to study outer automorphisms of the free group, and how we can extend this geometric tool to study outer automorphisms of RAAGs. Fasten your seatbelts, we are going to explore Outer Space.
Talk 1: Rachael Boyd
Title: Coxeter and Art...in groups
Abstract: I will introduce Coxeter and Artin groups with some historical context, and then focus on Right Angled Artin Groups (RAAGs). I will define the Salvetti complex for a given RAAG and show that it is a classifying space for the group using the theory of CAT(0) cube complexes.
Talk 2: Jean Pierre Mutanguha
Title: From flat doughnuts to outer space: a classifying space odyssey
Abstract: Starting with a motivating example of the moduli space of flat tori, we will define the outer space of a free group F. This can be thought of as a universal cover of a moduli space of graphs. In fact, we can do even better and view the moduli space of graphs as a (kind of) classifying space for Out(F), the outer automorphism group of F. We will deduce cohomological properties of Out(F) and also study individual elements of Out(F) by looking at their dynamics on outer space.
Talk 3: Corey Bregman
Title: Don't Panic and Always Carry a RAAG
Abstract: In this talk, we construct an outer space for right-angled Artin groups (RAAGs) generalising that of free groups. To do this, we will describe a family of locally CAT(0) spaces built from parallelotopes, whose fundamental group is a given RAAG. Moving around in the moduli space of all such metrics, we will be able to realize any automorphism, and thus obtain an action of the outer automorphism group of the RAAG. We show that the moduli space is finite dimensional, contractible and that the action is proper. Time permitting, we will outline recent work studying the fixed sets of finite subgroups, and discuss open problems. This is joint work with Ruth Charney and Karen Vogtmann.