Bounding the degree of generic sharp transitivity
Tuna Altınel, Joshua Wiscons
TL;DR
This work bounds the degree of generic sharp transitivity for permutation groups of finite Morley rank: if a generically sharply $t$-transitive action on a set of Morley rank $r$ has a pointwise stabilizer of a generic $(t-1)$-tuple that is an $L$-group, then $t \le r+2$, advancing the Borovik–Cherlin conjecture. The authors develop three independent tools: (i) a representation-theoretic analysis of Alt$(n)$/Sym$(n)$ actions on $L$-groups, (ii) a structural reduction for generically $2$-transitive actions with abelian stabilizers, and (iii) a detailed study of simple groups of Morley rank $6$. These ingredients allow the main bound to hold in broad settings, including $r \le 5$, even-type groups, and when stabilizers are solvable, and they illuminate the structure of the extremal case $t = r+2$. The work connects model-theoretic symmetry phenomena with algebraic-group techniques, contributing to the Morley-rank analogue of the PGL classification and providing new methods for analyzing high-degree generic transitivity. Overall, it narrows the landscape of possible generically sharply transitive actions and strengthens the bridge between finite Morley rank theory and classical algebraic group actions.
Abstract
We show that a generically sharply $t$-transitive permutation group of finite Morley rank on a set of rank $r$ satisfies $t\le r+2$ provided the pointwise stabilizer of a generic $(t-1)$-tuple is an $L$-group, which holds, for example, when this stabilizer is solvable or when $r\le 5$. This makes progress on the Borovik-Cherlin conjecture that every generically $(r+2)$-transitive permutation group of finite Morley rank on a set of rank $r$ is of the form $\operatorname{PGL}_{r+1}(F)$ acting naturally on $\mathbb{P}^r(F)$. Our proof is assembled from three key ingredients that are independent of the main theorem - these address actions of $\operatorname{Alt}(n)$ on $L$-groups of finite Morley rank, generically $2$-transitive actions with abelian point stabilizers, and simple groups of rank $6$.