2006/04/28 by Lucy Lifschitz, Dave Witte Morris, Lifschitz, Lucy +1
Mathematics · #20F60 #22E40 #57S25 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #math.GR #math.GT #msc:20F60 #msc:22E40 #msc:57S25
paper · pdf · doi:10.48550/arxiv.math/0604612
28 pages, no figures. In addition to minor corrections, one result was simplified (and strengthened) because of a stronger result in the final version of a joint paper with V.Chernousov
openalex publication_date 2006/04/28 · arxiv created 2007/06/19 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.