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