Vladimir Chernousov, Lucy Lifschitz, and Dave Witte Morris: Almost-minimal nonuniform lattices of higher rank

chernous@math.ualberta.ca, LLifschitz@math.ou.edu, Dave.Morris@uleth.ca

Submission: 2007, Sep 6

If G is any isotropic, almost simple algebraic group over an algebraic number field F, we show that G contains an isotropic, almost simple F-subgroup H, such that H is quasisplit, and the F_v-rank of H is greater than 1 at any archimedean valuation v of F for which the F_v-rank of G is greater than 1. This implies that if Gamma is a nonuniform, irreducible lattice in a semisimple Lie group whose real rank is greater than 1, then Gamma contains a subgroup that is isomorphic to a nonuniform, irreducible lattice in either SL(3,R), SL (3,C), or a direct product SL(2,R)^m x SL(2,C)^n, with m + n > 1. (In geometric terms, this can be interpreted as a statement about the existence of totally geodesic subspaces of finite-volume, noncompact, locally symmetric spaces of higher rank.)

2000 Mathematics Subject Classification:

Keywords and Phrases:

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