Random Permutations from Free Products

Let $\Gamma=G_1*G_2*\dots *G_r$ be a free product of a finite number of finite groups and a finite number of copies of the infinite cyclic group. We sample uniformly at random an action of $\Gamma$ on $N$ elements. In this talk, we will discuss a few tools we developed to help answer some natural questions involving the configuration described above, such as: For $\gamma\in \Gamma$, what is the expected number of fixed points of $\gamma$ in the action we sampled? What is the the typical behavior of the cycle structure of the permutation corresponding to $\gamma$ etc.

This is a joint with Doron Puder.