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On finite groups acting on ℤ2-homology 3-spheres

On finite groups acting on ℤ2-homology 3-spheres We consider finite groups G admitting orientation-preserving actions on homology 3-spheres (arbitrary, i.e. not necessarily free actions), concentrating on the case of nonsolvable groups. It is known that every finite group G admits actions on rational homology 3-spheres (and even free actions). On the other hand, the class of groups admitting actions on integer homology 3-spheres is very restricted (and close to the class of finite subgroups of the orthogonal group SO(4), acting on the 3-sphere). In the present paper, we consider the intermediate case of ℤ2-homology 3-spheres (i.e., with the ℤ2-homology of the 3-sphere where ℤ2 denote the integers mod two; we note that these occur much more frequently in 3-dimensional topology than the integer ones). Our main result is a list of finite nonsolvable groups G which are the candidates for orientation-preserving actions on ℤ2-homology 3-spheres. From this we deduce a corresponding list for the case of integer homology 3-spheres. In the integer case, the groups of the list are closely related to the dodecahedral group [InlineMediaObject not available: see fulltext.] or the binary dodecahedral group [InlineMediaObject not available: see fulltext.] most of these groups are subgroups of the orthogonal group SO(4) and hence admit actions on S 3. Roughly, in the case of ℤ2-homology 3-spheres the groups PSL(2,5) and SL(2,5) get replaced by the groups PSL(2,q) and SL(2,q), for an arbitrary odd prime power q. We have many examples of actions of the groups PSL(2,q) and SL(2,q) on ℤ2-homology 3-spheres, for various small values of q (constructed as regular coverings of suitable hyperbolic 3-orbifolds and 3-manifolds, using computer-supported methods to calculate the homology of the coverings). We think that all of them occur but have no method to prove this at present (in particular, the exact classification of the finite nonsolvable groups admitting actions on ℤ2-homology 3-spheres remains still open). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematische Zeitschrift Springer Journals

On finite groups acting on ℤ2-homology 3-spheres

Mecchia, Mattia; Zimmermann, Bruno
Mathematische Zeitschrift , Volume 248 (4) – May 5, 2004
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References (22)

Publisher
Springer Journals
Copyright
Copyright © 2004 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Mathematics, general
ISSN
0025-5874
eISSN
1432-1823
DOI
10.1007/s00209-004-0672-x
Publisher site
See Article on Publisher Site

Abstract

We consider finite groups G admitting orientation-preserving actions on homology 3-spheres (arbitrary, i.e. not necessarily free actions), concentrating on the case of nonsolvable groups. It is known that every finite group G admits actions on rational homology 3-spheres (and even free actions). On the other hand, the class of groups admitting actions on integer homology 3-spheres is very restricted (and close to the class of finite subgroups of the orthogonal group SO(4), acting on the 3-sphere). In the present paper, we consider the intermediate case of ℤ2-homology 3-spheres (i.e., with the ℤ2-homology of the 3-sphere where ℤ2 denote the integers mod two; we note that these occur much more frequently in 3-dimensional topology than the integer ones). Our main result is a list of finite nonsolvable groups G which are the candidates for orientation-preserving actions on ℤ2-homology 3-spheres. From this we deduce a corresponding list for the case of integer homology 3-spheres. In the integer case, the groups of the list are closely related to the dodecahedral group [InlineMediaObject not available: see fulltext.] or the binary dodecahedral group [InlineMediaObject not available: see fulltext.] most of these groups are subgroups of the orthogonal group SO(4) and hence admit actions on S 3. Roughly, in the case of ℤ2-homology 3-spheres the groups PSL(2,5) and SL(2,5) get replaced by the groups PSL(2,q) and SL(2,q), for an arbitrary odd prime power q. We have many examples of actions of the groups PSL(2,q) and SL(2,q) on ℤ2-homology 3-spheres, for various small values of q (constructed as regular coverings of suitable hyperbolic 3-orbifolds and 3-manifolds, using computer-supported methods to calculate the homology of the coverings). We think that all of them occur but have no method to prove this at present (in particular, the exact classification of the finite nonsolvable groups admitting actions on ℤ2-homology 3-spheres remains still open).

Journal

Mathematische Zeitschrift – Springer Journals

Published: May 5, 2004

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