Colloquium: Andrew Sale (Cornell U. )

March 1, 2018 @ 4:00 pm – 5:00 pm

Speaker: Andrew Sale (Cornell U.)

Title: On the outer automorphism groups of right-angled Artin and Coxeter groups

Abstract: In geometric group theory, a fundamental, and broad, question to answer is that of understanding the world of finitely presented groups. Two of the simplest examples are free groups Fn and free abelian groups Z^n. With Fn and Z^n being their extreme examples, right-angled Artin groups (RAAGs) give us some idea of what happens “between” these groups. RAAGs are an important class of groups which appear in diverse situations, perhaps most significantly in Agol’s proof of the virtual Haken conjecture.

In studying their outer automorphism groups, we are looking at a class of groups that again interpolates between two classically important families of groups: Out(Fn), the outer automorphism group of Fn, and GL(n,Z). While there are numerous similarities between these families, they also differ in some important contexts. One such context concerns the nature of quotients that they have, and I will describe a couple of properties that make rigorous the notion of “having many quotients”. I will explain what happens for outer automorphism groups of RAAGs, and also the closely related family of right-angled Coxeter groups, and the consequences this has for Kazhdan’s Property (T).