arXiv subjects

Lucy Lifschitz

Publications and source records attributed to Lucy Lifschitz.

4 recordsLinked to original sources

arXiv2007

Almost-minimal nonuniform lattices of higher rank

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.

math.GR↗

arXiv2007

Bounded generation and lattices that cannot act on the line

Let D be an irreducible lattice in a connected, semisimple Lie group G with finite center. Assume that the real rank of G is at least two, that G/D is not compact, and that G has more than one noncompact simple factor. We show that D has no orientation-preserving actions on the real line. (In algebraic terms, this means that D is not right orderable.) Under the additional assumption that no simple factor of G is isogenous to SL(2,R), applying a theorem of E.Ghys yields the conclusion that any orientation-preserving action of D on the circle must factor through a finite, abelian quotient of D. The proof relies on the fact, proved by D.Carter, G.Keller, and E.Paige, that SL(2,A) is boundedly generated by unipotents whenever A is a ring of integers with infinitely many units. The assumption that G has more than one noncompact simple factor can be eliminated if all noncocompact lattices in SL(3,R) and SL(3,C) are virtually boundedly generated by unipotents.

math.GR↗

arXiv2004

Isotropic nonarchimedean S-arithmetic groups are not left orderable

Let A = Z[c], where c is an irrational number whose square is rational, or let A = Z[1/r], where r > 1 is a square-free natural number. We show that no finite-index subgroup of SL(2,A) is left orderable. (Equivalently, these subgroups have no nontrivial orientation-preserving actions on the real line.) This implies that if G is an isotropic F-simple algebraic group over an algebraic number field F, then no nonarchimedean S-arithmetic subgroup of G is left orderable. Our proofs are based on the fact, proved by B.Liehl, that every element of SL(2,A) is a product of a bounded number of elementary matrices.

math.GR↗

arXiv2000

On automorphisms of arithmetic subgroups of unipotent groups in positive characteristic

Let F be a local field of positive characteristic, and let G be either a Heisenberg group over F, or a certain (nonabelian) two-dimensional unipotent group over F. If H is an arithmetic subgroup of G, we provide an explicit description of every automorphism of H. From this description, it follows that every automorphism of H virtually extends to a virtual automorphism of G.

math.GR↗