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

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.