Towards the recognition of $\operatorname{PGL}_n$ via a high degree of generic transitivity

TL;DR

This paper proves a conjecture about the maximum degree of generic transitivity in infinite permutation groups of finite Morley rank, specifically characterizing when the group is isomorphic to projective linear groups.

Contribution

It confirms the Borovik-Cherlin conjecture under additional 2-transitivity and quotient conditions, advancing the understanding of permutation groups of finite Morley rank.

Findings

Bound on generic transitivity degree established

Characterization of groups as projective linear groups

Inductive approach based on geometric quotient construction

Abstract

In 2008, Borovik and Cherlin posed the problem of showing that the degree of generic transitivity of an infinite permutation group of finite Morley rank $(X,G)$ is at most $n+2$ where $n$ is the Morley rank of $X$. Moreover, they conjectured that the bound is only achieved (assuming transitivity) by $\operatorname{PGL}_{n+1}(\mathbb{F})$ acting naturally on projective $n$-space. We solve the problem under the two additional hypotheses that (1) $(X,G)$ is $2$-transitive, and (2) $(X-\{x\},G_x)$ has a definable quotient equivalent to $(\mathbb{P}^{n-1}(\mathbb{F}),\operatorname{PGL}_{n}(\mathbb{F}))$. The latter hypothesis drives the construction of the underlying projective geometry and is at the heart of an inductive approach to the main problem.