# Actions on sets of Morley rank $2$

## Joshua Wiscons

### Hamilton College

Recently, Borovik and Cherlin initiated a broad study of permutation groups of finite Morley rank with a key topic being high degrees of generic transitivity. One of the main problems that they pose is to show that there is a natural upper bound on the degree of generic transitivity that depends only upon the rank of the set being acted on. Specifically, the problem is to show that the only groups of finite Morley rank with a generically $(n+2)$-transitive action on a set of rank $n$ are those of the form ${PGL}_{n+1}$. A solution when $n=1$, due to Hrushovski, has been known for a few decades as in this case the set is strongly minimal. In this talk, I will present recent work, joint with Tuna Altinel, addressing the case of $n=2$. The analysis of these actions makes considerable use of the structure of groups of small rank, and as such, I will also discuss some new results on groups of Morley rank $4$.