Extending finite group actions from surfaces to handlebodies

Reni, Marco; Zimmermann, Bruno

doi:10.1090/S0002-9939-96-03515-0

Extending finite group actions from surfaces to handlebodies
HTML articles powered by AMS MathViewer

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

PDF | Request permission

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

A. L. Edmonds and J. H. Ewing, Remarks on the cobordism group of surface diffeomorphisms, Math. Ann. 259 (1982), no. 4, 497–504. MR 660044, DOI 10.1007/BF01466055
Monique Gradolato and Bruno Zimmermann, Extending finite group actions on surfaces to hyperbolic $3$-manifolds, Math. Proc. Cambridge Philos. Soc. 117 (1995), no. 1, 137–151. MR 1297900, DOI 10.1017/S0305004100072960
Henry Glover and Denis Sjerve, The genus of $\textrm {PSl}_2(q)$, J. Reine Angew. Math. 380 (1987), 59–86. MR 916200, DOI 10.1515/crll.1987.380.59
André Haefliger, Groupoïdes d’holonomie et classifiants, Astérisque 116 (1984), 70–97 (French). Transversal structure of foliations (Toulouse, 1982). MR 755163
Rubén A. Hidalgo, On Schottky groups with automorphisms, Ann. Acad. Sci. Fenn. Ser. A I Math. 19 (1994), no. 2, 259–289. MR 1274083
Darryl McCullough, Andy Miller, and Bruno Zimmermann, Group actions on handlebodies, Proc. London Math. Soc. (3) 59 (1989), no. 2, 373–416. MR 1004434, DOI 10.1112/plms/s3-59.2.373
W. H. Meeks, III and S.-T. Yau, The equivariant loop theorem for three-dimensional manifolds, The Smith Conjecture, Academic Press, New York, 1984, pp. 153–163.
Marco Reni and Bruno Zimmermann, Handlebody orbifolds and Schottky uniformizations of hyperbolic $2$-orbifolds, Proc. Amer. Math. Soc. 123 (1995), no. 12, 3907–3914. MR 1307560, DOI 10.1090/S0002-9939-1995-1307560-5
Jean-Pierre Serre, Trees, Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell. MR 607504, DOI 10.1007/978-3-642-61856-7
W. Thurston, The geometry and topology of three-dimensional manifolds, Lecture Notes, Princeton University, 1978.
—, Three-manifolds with symmetry, Preprint, 1982.
Heiner Zieschang, Elmar Vogt, and Hans-Dieter Coldewey, Surfaces and planar discontinuous groups, Lecture Notes in Mathematics, vol. 835, Springer, Berlin, 1980. Translated from the German by John Stillwell. MR 606743, DOI 10.1007/BFb0089692
Bruno Zimmermann, Surfaces and the second homology of a group, Monatsh. Math. 104 (1987), no. 3, 247–253. MR 918475, DOI 10.1007/BF01547955
Bruno Zimmermann, Finite abelian group actions on surfaces, Yokohama Math. J. 38 (1990), no. 1, 13–21. MR 1093659
Bruno Zimmermann, Generators and relations for discontinuous groups, Generators and relations in groups and geometries (Lucca, 1990) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 333, Kluwer Acad. Publ., Dordrecht, 1991, pp. 407–436. MR 1206922, DOI 10.1007/978-94-011-3382-1_{1}4