Abstract
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.
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Communicated by A. Constantin.
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Zimmermann, B.P. On large orientation-reversing finite group-actions on 3-manifolds and equivariant Heegaard decompositions. Monatsh Math 191, 437–444 (2020). https://doi.org/10.1007/s00605-019-01303-8
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DOI: https://doi.org/10.1007/s00605-019-01303-8
Keywords
- 3-Manifold
- Finite group-action
- Equivariant Heegaard decomposition
- Maximally symmetric 3-manifold
- Tetrahedral Coxeter group