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

TL;DR

This paper establishes a quadratic upper bound on the order of finite groups acting faithfully on connected sums of S^2 imes S^1, developing a new calculus for such actions using handle-orbifolds and graphs of groups.

Contribution

It introduces a novel calculus for finite group actions on H_g, proving a quadratic bound on group order and establishing a Jordan-type bound for these actions.

Findings

Finite groups acting on H_g have order bounded quadratically in g.

No linear bound exists for the order of such groups.

Develops a handle-orbifold and graph-based framework for analyzing group actions.

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 that 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.