Bounded generation and lattices that cannot act on the line
摘要
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.
引用
@article{arxiv.math/0604612,
title = {Bounded generation and lattices that cannot act on the line},
author = {Lucy Lifschitz and Dave Witte Morris},
journal= {arXiv preprint arXiv:math/0604612},
year = {2007}
}
备注
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