Extending finite group actions from surfaces to handlebodies
Extending finite group actions from surfaces to handlebodies
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将有限群动作从曲面扩展到手柄
DOI：
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发表时间：
1996
期刊：
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通讯作者：
B. Zimmermann
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文献类型：
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作者：
Marco Reni;B. Zimmermann
我们证明了有限二面体群在封闭可定向曲面F上的每个作用都延伸到三维柄体V，其中V = F。在有限阿贝尔群G的情形下，我们给出曲面上的G-作用扩张到紧致3-流形，或者等价地，在这种情形下，扩张到3-维拟体的充要条件;特别地，有限阿贝尔群的所有（不动点）自由作用都扩张到拟体。这不再是真正的自由行动的任意有限群：我们给出了一个程序，使我们能够构建自由行动的有限群表面不延伸到一个cross-body。我们还证明了PSL（2，27）在亏格为g = 118的曲面F上的唯一的84（g−1）阶Hurwitz作用并不延伸到任何紧致3-流形M，其中M = F，从而解决了PSL（2，q）型的唯一低阶Hurwitz作用的情况，这在以前的论文中仍然是公开的（Math. Proc.剑桥哲学学会论文集，117（1995），137-151）。
We show that every action of a finite dihedral group on a closed orientable surface F extends to a 3-dimensional handlebody V , with ∂V = 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 F of genus g = 118 does not extend to any compact 3-manifold M with ∂M = 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).