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On finite groups acting on a connected sum of 3-manifolds $S^2\times S^1$

Volume 226 / 2014

Bruno P. Zimmermann Fundamenta Mathematicae 226 (2014), 131-142 MSC: 57M60, 57M17, 57S25. DOI: 10.4064/fm226-2-3

Abstract

Let $H_g$ denote 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 prove that, for a finite group $G$ acting on $H_g$ which induces a faithful action on the fundamental group, there is an upper bound for the order of $G$ which is quadratic in $g$, but there does not exist a linear bound in $g$. This implies then a Jordan-type bound for arbitrary finite group actions on $H_g$ which is quadratic in $g$. For the proofs we develop a calculus for finite group actions on $H_g$, by codifying such actions by handle-orbifolds and finite graphs of finite groups.

Authors

  • Bruno P. ZimmermannDipartimento di Matematica e Geoscienze
    Università degli Studi di Trieste
    34127 Trieste, Italy
    e-mail

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