# Geometric models for classifying spaces of $GLn(Fq)$.

The title pretty much says it. In a follow-up to my question about alternating groups, does anyone know of a "geometric" model for $BGL_n(F_q)$? By "geometric" I mean "a space you would have heard about even if you aren't studying classifying spaces"; I mean to exclude general constructions such as standard nerve constructions, infinite joins, intractable quotients of frame bundles, etc.

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I have a couple thoughts about possible sources of such models. One is that sometimes models of BAut_C X where C is some category can be of the form "embeddings of X in a universal, contractible C-object." Examples include symmetric groups (BS_n = embeddings of points in R^\infty), general linear groups (linear embeddings of a vector space in a universal vector space (also works for G-actions)), and diffeomorphism groups (subsets of R^\infty diffeomorphic to a fixed manifold = embeddings modulo diffeomorphisms). Of course, universal F_q vector spaces are discrete... –  Dev Sinha Jul 31 '10 at 4:22
The other thought is that there might be a poset/ simplicial construction which is geometrically defined - say built from the poset of finite-dimensional subspaces of (F_q)^\infty under inclusion or perhaps from some matroid theory. –  Dev Sinha Jul 31 '10 at 5:20