Stable commutator lengths in right-angled Artin groups

Geometry Topology Seminar
Monday, March 14, 2016 - 14:05
1 hour (actually 50 minutes)
Skiles 006
U Oklahoma
The commutator length of an element g in the commutator subgroup [G,G] of agroup G is the smallest k such that g is the product of k commutators. WhenG is the fundamental group of a topological space, then the commutatorlength of g is the smallest genus of a surface bounding a homologicallytrivial loop that represents g. Commutator lengths are notoriouslydifficult to compute in practice. Therefore, one can ask for asymptotics.This leads to the notion of stable commutator length (scl) which is thespeed of growth of the commutator length of powers of g. It is known thatfor n > 2, SL(n,Z) is uniformly perfect; that is, every element is aproduct of a bounded number of commutators, and hence scl is 0 on allelements. In contrast, most elements in SL(2,Z) have positive scl. This isrelated to the fact that SL(2,Z) acts naturally on a tree (its Bass-Serretree) and hence has lots of nontrivial quasimorphisms.In this talk, I will discuss a result on the stable commutator lengths inright-angled Artin groups. This is a broad family of groups that includesfree and free abelian groups. These groups are appealing to work withbecause of their geometry; in particular, each right-angled Artin groupadmits a natural action on a CAT(0) cube complex. Our main result is anexplicit uniform lower bound for scl of any nontrivial element in anyright-angled Artin group. This work is joint with Talia Fernos and MaxForester.