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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Extending finite group actions
from surfaces to handlebodies


Authors: Marco Reni and Bruno Zimmermann
Journal: Proc. Amer. Math. Soc. 124 (1996), 2877-2887
MSC (1991): Primary 57M60; Secondary 57S25, 30F99
MathSciNet review: 1343720
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Abstract: We show that every action of a finite dihedral group on a closed orientable surface $\mathcal F$ extends to a 3-dimensional handlebody $\mathcal V$, with $\partial \mathcal V=\mathcal F$. In the case of a finite abelian group $G$, we give necessary and sufficient conditions for a $G$-action on a surface to extend to a compact $3$-manifold, or, equivalently in this case, to a 3-dimensional handlebody; in particular all (fixed-point) free actions of finite abelian groups extend to handlebodies. This is no longer true for free actions of arbitrary finite groups: we give a procedure which allows us to construct free actions of finite groups on surfaces which do not extend to a handlebody. We also show that the unique Hurwitz action of order $84(g-1)$ of $PSL(2,27)$ on a surface $\mathcal F$ of genus $g=118$ does not extend to any compact 3-manifold $M$ with $\partial M=\mathcal F$, thus resolving the only case of Hurwitz actions of type $PSL(2,q)$ of low order which remained open in an earlier paper (Math. Proc. Cambridge Philos. Soc. 117 (1995), 137--151).


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Additional Information

Marco Reni
Affiliation: Dipartimento di Scienze Matematiche, Università degli Studi di Trieste, 34100 Trieste, Italy
Email: reni@univ.trieste.it

Bruno Zimmermann
Affiliation: Dipartimento di Scienze Matematiche, Università degli Studi di Trieste, 34100 Trieste, Italy
Email: zimmer@univ.trieste.it

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03515-0
PII: S 0002-9939(96)03515-0
Keywords: Surface, handlebody, handlebody orbifold, finite group action
Received by editor(s): February 2, 1995
Communicated by: Ronald Stern
Article copyright: © Copyright 1996 American Mathematical Society