Almost-minimal nonuniform lattices of higher rank

TL;DR

This paper proves that nonuniform lattices in higher rank semisimple Lie groups contain subgroups isomorphic to certain lower-rank lattices, revealing structural properties of these complex geometric and algebraic objects.

Contribution

It establishes the existence of specific lower-rank nonuniform lattices within higher-rank lattices, advancing understanding of their subgroup structure and geometric implications.

Findings

Existence of subgroups isomorphic to SL(3,R), SL(3,C), or products of SL(2,R) and SL(2,C)

Implication for totally geodesic subspaces in locally symmetric spaces

Identification of certain Q-subgroups with higher real rank

Abstract

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.