Consider the following example: Take a random permutation on $N$ elements denoted by $A$, which is drawn uniformly at random from the collection $\{ \sigma \in S_{N} \mid \sigma^3 = {id} \}$. Additionally, let $B$ be another random permutation, uniformly selected from the set $\{ \sigma \in S_{N} \mid \sigma^2 = {id} \}$. Let $\gamma$ be a word in $A$ and $B$; for instance, $\gamma$ could be represented as $ABAAB$. This way $\gamma$ defines a new random permutation, given by choosing $A$ and $B$ and then take the composition $A\circ B \circ A \circ A \circ B$.
In the forthcoming talk, we will discuss several results regarding the distribution of such a $\gamma$-random permutation as $N$ grows to infinity. For instance, we present the calculation of the limit of the expected number of fixed points of a $\gamma$-random permutation. We show a surprising relation between such outcomes to topological and algebraic invariants of elements in free products of groups.
This talk is based on my joint work with Doron Puder from Tel Aviv University.