On finite groups acting on a connected sum $\sharp_g (S^2 \times S^1)$

Details On finite groups acting on a connected sum $\sharp_g (S^2 \times S^1)$ On finite groups acting on a connected sum $\sharp_g (S^2 \times S^1)$

2012-02-24

arxiv.org/abs/1202.5427v1

Mathematics

Geometric Topology

9 pages

Scientific paper

Let $H_g = \sharp_g (S^2 \times S^1)$ be the closed 3-manifold obtained as the connected sum of $g$ copies of $S^2 \times S^1$, with free fundamental group of rank $g$. We present a calculus for finite group actions on $H_g$, by codifying such actions by handle orbifolds and finite graphs of finite groups. As an application (which in fact motivated the presentation) we prove that, for a finite cyclic group $G$ acting on $H_g$ such that the induced action is faithful on the fundamental group, there is an upper bound for the order of $G$ which is quadratic in $g$, but that there does not exist a linear bound. We note that this implies a cubic bound for the orders of arbitrary finite groups $G$ but believe that there exists a quadratic bound also in this case.

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