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

Bruno P. Zimmermann¹

¹ Dipartimento di Matematica e Geoscienze Università degli Studi di Trieste 34127 Trieste, Italy

Fundamenta Mathematicae, Tome 226 (2014) no. 2, pp. 131-142

Résumé

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.

Détail

DOI : 10.4064/fm226-2-3

Keywords: denote closed manifold obtained connected sum nbsp copies times fundamental group rank nbsp prove finite group acting which induces faithful action fundamental group there upper bound order nbsp which quadratic there does exist linear bound nbsp implies jordan type bound arbitrary finite group actions which quadratic proofs develop calculus finite group actions codifying actions handle orbifolds finite graphs finite groups