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On the first cohomology of arithmetic groups
Authors:C S Rajan  T N Venkataramana
Institution:(1) School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay - 400 005, India. e-mail: rajan@math.tifr.res.in; venky@math.tifr.res.in, IN
Abstract:We study the restriction to smaller subgroups, of cohomology classes on arithmetic groups (possibly after moving the class by Hecke correspondences), especially in the context of first cohomology of arithmetic groups. We obtain vanishing results for the first cohomology of cocompact arithmetic lattices in SU(n,1) which arise from hermitian forms over division algebras D of degree p 2, p an odd prime, equipped with an involution of the second kind. We show that it is not possible for a ‘naive’ restriction of cohomology to be injective in general. We also establish that the restriction map is injective at the level of first cohomology for non co-compact lattices, extending a result of Raghunathan and Venkataramana for co-compact lattices. Received: 14 September 2000 / Accepted: 6 June 2001
Keywords:Mathematics Subject Classification (2000): Primary 11F75  Secondary 22E40  22E41
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