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Multiplicative Invariants and the Finite Co-Hopfian Property

Part of: Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  21 December 2009

J. J. A. M. Humphreys  and
F. E. A. Johnson
J. J. A. M. Humphreys
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, U.K., E-mail: jodie.humphreys@cantab.net
F. E. A. Johnson
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, U.K., E-mail: feaj@math.ucl.ac.uk
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Abstract

A group is said to be, finitely co-Hopfian when it contains no proper subgroup of finite index isomorphic to itself. It is known that irreducible lattices in semisimple Lie groups are finitely co-Hopfian. However, it is not clear, and does not appear to be known, whether this property is preserved under direct product. We consider a strengthening of the finite co-Hopfian condition, namely the existence of a non-zero multiplicative invariant, and show that, under mild restrictions, this property is closed with respect to finite direct products. Since it is also closed with respect to commensurability, it follows that lattices in linear semisimple groups of general type are finitely co-Hopfian.

MSC classification

Secondary: 20E34: General structure theorems
Type
Research Article
Information
Mathematika , Volume 55 , Issue 1-2 , December 2009 , pp. 115 - 127
Copyright
Copyright © University College London 2009

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