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Title of Journal: Math Z

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Abbravation: Mathematische Zeitschrift

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Springer Berlin Heidelberg

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DOI

10.1016/0167-899x(85)90022-9

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1432-1823

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Actions of arithmetic groups on homology spheres a

Authors: Martin R Bridson Fritz Grunewald Karen Vogtmann
Publish Date: 2013/08/07
Volume: 276, Issue: 1-2, Pages: 387-395
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Abstract

We establish lower bounds on the dimensions in which arithmetic groups with torsion can act on acyclic manifolds and homology spheres The bounds rely on the existence of elementary pgroups in the groups concerned In some cases including mathrmSp2nmathbb Z the bounds we obtain are sharp if X is a generalized mathbb Z /3homology sphere of dimension less than 2n1 or a mathbb Z /3acyclic mathbb Z /3homology manifold of dimension less than 2n and if nge 3 then any action of mathrmSp2nmathbb Z by homeomorphisms on X is trivial if n=2 then every action of mathrmSp2nmathbb Z on X factors through the abelianization of mathrmSp4mathbb Z which is mathbb Z /2We thank Alex Lubotzky Gopal Prasad and Alan Reid for their helpful comments concerning the material in Sect 4 Most particularly we thank Dan Segal for his notes on this material from which we borrowed heavily We also thank the Institute MittagLeffler Djursholm Sweden for its hospitality during the preparation of this manuscript Tragically the second author did not survive to see this project completed He is sorely missed for many reasons Any deficiencies in the final version of this paper are the responsibility of the first and third authors alone

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