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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 $ {V_g}$ be an orientable three-dimensional handlebody with genus $ g > 1$. Let $ N(g)$ be the largest order among all finite groups which act effectively on $ {V_g}$ and preserve orientation. We show that $ 4(g + 1) \leq N(g) \leq 12(g - 1)$, and that $ N(g)$ equals either $ 8(q - 1)$ or $ 12(g - 1)$ when $ g$ is odd. Moreover each of the indicated upper and lower bounds are achieved for infinitely many genera $ g$. The techniques which are used lead to more detailed results and also specialize to yield similar results for compact surfaces with nonempty boundary.

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Keywords: $ 3$-dimensional handlebody, finite group action, graph of groups, compact $ 2$-manifold with boundary
Article copyright: © Copyright 1989 American Mathematical Society

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