The family of right-angled Artin groups (RAAGs) interpolates 
between free groups and free abelian groups. These groups are defined by 
a simplicial graph: the vertices correspond to generators, and two 
generators commute if and only if they are connected by an edge in the 
defining graph. A key feature of RAAGs is that many of their algebraic 
properties can be detected purely in terms of the combinatorics of the 
defining graph.
The family of outer automorphism groups of RAAGs similarly interpolates 
between Out(F_n) and GL(n, Z). While the l2-Betti numbers of GL(n, Z) 
are well understood, those of Out(F_n) remain largely mysterious. The 
aim of this talk is to introduce automorphism groups of RAAGs and to 
present a combinatorial criterion, expressed in terms of the defining 
graph, that characterizes when the first l2-Betti number of Out(RAAG) 
vanishes.
If time permits, we will also discuss higher l2-Betti numbers and 
algebraic fibring properties of these group

You can cut a cake in half, a pizza into slices, but can you cut an infinite group? I'll tell you about my sensei's secret cutting technique and demonstrate a couple of examples. There might be some spectral goodies at the end too.