Toward the recognition of 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 ${\text{PGL}}_{n+1}(\mathbb{F})$ 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 $({\mathbb{P}}^{n-1}(\mathbb{F}),{\text{PGL}}_n(\mathbb{F}))$. The latter hypothesis drives the construction of the underlying projective geometry and is at the heart of an inductive approach to the main problem.

Keywords:

• Multiple transitivity
• finite Morley rank
• projective geometry

2010 MATHEMATICS SUBJECT CLASSIFICATION::

• Primary 20B22
• Secondary 03C60

Acknowledgements

The second author would like to acknowledge the warm hospitality of Université Claude Bernard Lyon-1, where the majority of the work for this article was carried out, as well as support from Hamilton College and California State University, Sacramento (through the Research and Creative Activity Faculty Awards Program) for making the visits to Lyon possible. The authors would also like to warmly thank the anonymous referee for several helpful comments that served to improve the clarity of the article.

Additional information

Funding

J. Wiscons was partially supported by the Sacramento State Research and Creative Activity Faculty Awards Program.