Random Matrices From GLn(q) Sampled by Words

Every word in a free group induces a probability distribution on every finite group by substituting the letters of w by uniformly random elements of the group. The connection between such distributions on the symmetric group and the poset of finitely generated subgroups of the free group was established several years ago.

In a more recent project, joint with Danielle Ernst-West and Matan Seidel, we study word-distributions on matrix groups over finite fields. To our surprise, we discovered a beautiful connection with free group algebras, featuring interesting analogies with the case of the symmetric group.

The talk will include gentle introduction to free groups and to free group algebras. I will describe what we found, the analogies with the symmetric group case, as well as some intriguing conjectures