Groups Acting Generically Multiply Transitively on Solvable Groups

TL;DR

This paper classifies highly transitive actions of groups on solvable groups within the finite Morley rank framework, showing they are essentially linear actions on vector spaces over algebraically closed fields.

Contribution

It completes the classification of generically multiply transitive actions on solvable groups in finite Morley rank, extending previous results and applying to primitive groups of affine type.

Findings

Connected groups of finite Morley rank acting generically m-transitively are isomorphic to GL_m(F)

The solvable group V is a vector space over an algebraically closed field

The action is equivalent to the natural linear action of GL_m(F)

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 $\operatorname{rk}(X)+1$.