Bounded generation of S-arithmetic subgroups of isotropic orthogonal groups over number fields

2005/08/24 by Igor V. Erovenko, Erovenko, Igor V., Andrei S. Rapinchuk +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Finite Group Theory Research #math.GR

paper · pdf · doi:10.48550/arxiv.math/0508480

An extended version of the paper accepted by the Journal of Number Theory, it includes a self-contained proof of Witt's theorem for local lattices

arxiv created 2005/08/29 · arxiv updated 2009/12/01

Abstract

Let f be a nondegenerate quadratic form in at least 5 variables over a number field K and let S be a finite set of valuations of K containing all Archimedean ones. We prove that if the Witt index of f is at least 2 or it is 1 and S contains a non-Archimedean valuation, then the S-arithmetic subgroups of the special orthogonal group of f have bounded generation. These groups provide a series of examples of boundedly generated S-arithmetic groups in isotropic, but not quasi-split, algebraic groups.

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