The title pretty much says it.  In a follow-up to my [question about alternating groups][1], 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.


  [1]: http://mathoverflow.net/questions/30113/geometric-model-for-classifying-spaces-of-alternating-groups