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COMPLETE EMBEDDINGS OF GROUPS

Part of: Structure and classification of infinite or finite groups Special aspects of infinite or finite groups

Published online by Cambridge University Press:  26 January 2024

MARTIN R. BRIDSON Open the ORCID record for MARTIN R. BRIDSON [Opens in a new window]  and
HAMISH SHORT Open the ORCID record for HAMISH SHORT [Opens in a new window]
MARTIN R. BRIDSON*
Affiliation:
Mathematical Institute, Andrew Wiles Building, ROQ, Woodstock Road, Oxford OX2 6GG, UK
HAMISH SHORT
Affiliation:
L.A.T.P., U.M.R. 6632, C.M.I., 39 Rue Joliot–Curie, Université de Provence, F–13453 Marseille cedex 13, France e-mail: hamish@cmi.univ-mrs.fr
*
e-mail: bridson@maths.ox.ac.uk
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Abstract

Every countable group G can be embedded in a finitely generated group $G^*$ that is hopfian and complete, that is, $G^*$ has trivial centre and every epimorphism $G^*\to G^*$ is an inner automorphism. Every finite subgroup of $G^*$ is conjugate to a finite subgroup of G. If G has a finite presentation (respectively, a finite classifying space), then so does $G^*$. Our construction of $G^*$ relies on the existence of closed hyperbolic 3-manifolds that are asymmetric and non-Haken.

Keywords

complete groupsasymmetric hyperbolic manifoldsembedding

MSC classification

Primary: 20F65: Geometric group theory
Secondary: 20E08: Groups acting on trees 20F67: Hyperbolic groups and nonpositively curved groups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 9
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

For our friend and coauthor Chuck Miller in his ninth decade, with deep respect and affection

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