Large groups of symmetries of handlebodies

Authors:
A. Miller and B. Zimmermann

Journal:
Proc. Amer. Math. Soc. **106** (1989), 829-838

MSC:
Primary 57M99; Secondary 57S25, 57S30

MathSciNet review:
962246

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Abstract: Let be an orientable three-dimensional handlebody with genus . Let be the largest order among all finite groups which act effectively on and preserve orientation. We show that , and that equals either or when is odd. Moreover each of the indicated upper and lower bounds are achieved for infinitely many genera . The techniques which are used lead to more detailed results and also specialize to yield similar results for compact surfaces with nonempty boundary.

**[A]**Robert D. M. Accola,*On the number of automorphisms of a closed Riemann surface*, Trans. Amer. Math. Soc.**131**(1968), 398–408. MR**0222281**, 10.1090/S0002-9947-1968-0222281-6**[GM]**Newcomb Greenleaf and Coy L. May,*Bordered Klein surfaces with maximal symmetry*, Trans. Amer. Math. Soc.**274**(1982), no. 1, 265–283. MR**670931**, 10.1090/S0002-9947-1982-0670931-X**[H]**W. J. Harvey,*Cyclic groups of automorphisms of a compact Riemann surface*, Quart. J. Math. Oxford Ser. (2)**17**(1966), 86–97. MR**0201629****[M]**C. Maclachlan,*A bound for the number of automorphisms of a compact Riemann surface.*, J. London Math. Soc.**44**(1969), 265–272. MR**0236378****[M]**Coy L. May,*A bound for the number of automorphisms of a compact Klein surface with boundary*, Proc. Amer. Math. Soc.**63**(1977), no. 2, 273–280. MR**0435385**, 10.1090/S0002-9939-1977-0435385-0**[MMZ]**D. McCullough, A. Miller, and B. Zimmermann,*Group actions on handlebodies*, Proc. London Math. Soc. (to appear).**[MMZ]**-,*Group actions on nonclosed surfaces*, 1988, (preprint).**[ZVC]**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****[Z]**Bruno Zimmermann,*Über Abbildungsklassen von Henkelkörpern*, Arch. Math. (Basel)**33**(1979/80), no. 4, 379–382 (German). MR**564296**, 10.1007/BF01222772**[Z]**Bruno Zimmermann,*Über Homöomorphismen 𝑛-dimensionaler Henkelkörper und endliche Erweiterungen von Schottky-Gruppen*, Comment. Math. Helv.**56**(1981), no. 3, 474–486 (German). MR**639363**, 10.1007/BF02566224**[Z]**Reza Zomorrodian,*Nilpotent automorphism groups of Riemann surfaces*, Trans. Amer. Math. Soc.**288**(1985), no. 1, 241–255. MR**773059**, 10.1090/S0002-9947-1985-0773059-6**[Z]**Reza Zomorrodian,*Classification of 𝑝-groups of automorphisms of Riemann surfaces and their lower central series*, Glasgow Math. J.**29**(1987), no. 2, 237–244. MR**901670**, 10.1017/S0017089500006881

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1989-0962246-9

Keywords:
-dimensional handlebody,
finite group action,
graph of groups,
compact -manifold with boundary

Article copyright:
© Copyright 1989
American Mathematical Society