On large orientation-reversing finite group-actions on 3-manifolds and equivariant Heegaard decompositions

We consider finite group-actions on closed, orientable and nonorientable 3-manifolds; such a finite group-action leaves invariant the two handlebodies of a Heegaard splitting of M of some genus g . The maximal possible order of a finite group-action of an orientable or nonorientable handlebody of ge...

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Bibliographic Details
Published in:Monatshefte für Mathematik 2020-02, Vol.191 (2), p.437-444
Main Author: Zimmermann, Bruno P.
Format: Article
Language:English
Subjects:
Euclidean geometry
Invariants
Manifolds
Mathematics
Mathematics and Statistics
Quotients
Splitting
Subgroups
Tetrahedra
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Summary:We consider finite group-actions on closed, orientable and nonorientable 3-manifolds; such a finite group-action leaves invariant the two handlebodies of a Heegaard splitting of M of some genus g . The maximal possible order of a finite group-action of an orientable or nonorientable handlebody of genus g > 1 is 24 ( g - 1 ) , and in the present paper we characterize the 3-manifolds M and groups G for which the maximal possible order | G | = 24 ( g - 1 ) is obtained, for some G -invariant Heegaard splitting of genus g > 1 . If M is reducible then it is obtained by doubling an action of maximal possible order 24 ( g - 1 ) on a handlebody of genus g . If M is irreducible then it is a spherical, Euclidean or hyperbolic manifold obtained as a quotient of one of the three geometries by a normal subgroup of finite index of a Coxeter group associated to a Coxeter tetrahedron, or of a twisted version of such a Coxeter group.
ISSN:0026-9255
1436-5081
DOI:10.1007/s00605-019-01303-8