Extending finite group actions from surfaces to handlebodies
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- by Marco Reni and Bruno Zimmermann
- Proc. Amer. Math. Soc. 124 (1996), 2877-2887
- DOI: https://doi.org/10.1090/S0002-9939-96-03515-0
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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).References
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Bibliographic 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
- Received by editor(s): February 2, 1995
- Communicated by: Ronald Stern
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 2877-2887
- MSC (1991): Primary 57M60; Secondary 57S25, 30F99
- DOI: https://doi.org/10.1090/S0002-9939-96-03515-0
- MathSciNet review: 1343720