Groups Acting Generically Multiply Transitively on Solvable Groups

Abstract

In this work, we complete the classification of generically multiply transitive actions of groups on solvable groups in the finite Morley rank setting. We prove that if $G$ is a connected group of finite Morley rank acting definably, faithfully and generically $m$-transitively on a connected solvable group $V$ of finite Morley rank where $\operatorname{rk}(V)\leqslant m$, then $\operatorname{rk}(V)=m$, $V$ is a vector space of dimension $m$ over an algebraically closed field $F$, $G\cong \operatorname{GL}_m(F)$, and the action is equivalent to the natural action of $\operatorname{GL}_m(F)$ on $F^m$. This generalises our previous work arXiv:2107.09997. As an application of our result, we classify definably primitive groups of finite Morley rank and affine type acting on a set $X$ with a generic transitivity degree of $\operatorname{rk}(X)+1$.