Almost-minimal nonuniform lattices of higher rank

Details Almost-minimal nonuniform lattices of higher rank Almost-minimal nonuniform lattices of higher rank

2007-05-30

arxiv.org/abs/0705.4330v3

Mathematics

Group Theory

23 pages. Minor corrections, and added remarks about which of the subgroups we construct are simply connected

Scientific paper

If Gamma is a nonuniform, irreducible lattice in a semisimple Lie group whose real rank is greater than 1, we show 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.) Another formulation of the result states that if G is any isotropic, almost simple algebraic group over Q (the rational numbers), such that the real rank of G is greater than 1, then G contains an isotropic, almost simple Q-subgroup H, such that H is quasisplit, and the real rank of H is greater than 1.

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