Colloquium: Tuesday, April 12, 2022. Speaker: Ariel Yadin (Ben-Gurion). Title: “Abundance of subgroups via random walk entropy”.
The talk (given in person in room (614) will also be broadcasted on zoom, and the link for the meetings is:
https://us02web.zoom.us/j/83337601824
Time: 14:00
Abstract: Given a (finitely generated) group, we are interested in investigating whether the group has a rich family of subgroups or not. This question is not precise enough, as stated. What we will focus on for this talk, is whether a group has many normal subgroups, or whether it has many invariant random subgroups (IRS), which are a probabilistic generalization of a normal subgroup. For example, a simple group does not have any non-trivial normal subgroups. A famous theorem of Margulis tells us that SL(3,Z) has “few” normal subgroups. Stuck & Zimmer extend this also to IRSs.
Contrary to the high rank SL(n,Z) case, we believe that SL(2,Z) should have many many normal subgroups and IRSs. Since SL(2,Z) contains a free group on 2 elements with finite index, this leads one to consider subgroups of finitely generated free groups.
Indeed, extending results of Bowen, in joint works with Y. Hartman and L. Ron-George, we show that finitely generated free groups and SL(2,Z) have an abundance of ergodic IRSs, where “abundant” here is measured using random walk entropy. Specifically, all a-priori random walk entropy values can be realized by some ergodic IRS. The question whether entropy values can be realized with normal subgroups is still open.
Our methods involve a new construction of IRSs, by gluing together Schreier graphs, and we combine algebraic constructions with geometric and random walk considerations.
No prior knowledge of the notions of IRS, random walk entropy, etc. is assumed.
