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Finite geometry after Aschbacher's Theorem: PGL(n, q) from a Kleinian viewpoint

Studying the geometry of a group G leads us to questions about its maximal subgroups and primitive permutation representations (the G-invariant relations and similar structures, the base size, recognition problems, and so on). Taking the point of view that nite projective geometry is the geometry of the groups PGL(n; q), Aschbacher's theorem gives us eight natural families of geometric objects, with greater or smaller degrees of familiarity. This paper presents some speculations on how the subject could develop from this point of view.