Extracting stable commutator length and other invariants of words from random matrices

Doron Puder (Tel Aviv and IAS)

Abstract: Every word w in a free group induces a natural probability measure on the unitary group of matrices. Somewhat surprisingly, these random matrices give rise to certain maps from surfaces to the bouquet, which are related, in turn, to the commutator length of w and, even more so, to its stable commutator length. Tweaking the random-matrix side of this story, we get a recipe for defining additional invariants of words. In the talk, which I aim to make very accessible, I will describe this interesting phenomena, focusing on the topological part of the story. This is based on joint works with Michael Magee and Yotam Shomroni.