Automatic continuity for groups whose torsion subgroups are small

TL;DR

The paper establishes conditions under which group homomorphisms from locally compact groups to discrete groups are continuous, especially when the target groups have small torsion subgroups, with applications to geometric group theory.

Contribution

It proves a new automatic continuity result for homomorphisms into groups with small torsion subgroups, extending previous understanding in topological group theory.

Findings

Homomorphisms are continuous unless they factor through a torsion subgroup

Surjective homomorphisms with no non-trivial normal torsion subgroups are automatically continuous

Applications include continuity results for automorphism groups of right-angled Artin groups and Helly groups

Abstract

We prove that a group homomorphism $\varphi\colon L\to G$ from a locally compact Hausdorff group $L$ into a discrete group $G$ either is continuous, or there exists a normal open subgroup $N\subseteq L$ such that $\varphi(N)$ is a torsion group provided that $G$ does not include $\mathbb{Q}$ or the $p$-adic integers $\mathbb{Z}_p$ or the Pr\"ufer $p$-group $\mathbb{Z}(p^\infty)$ for any prime $p$ as a subgroup, and if the torsion subgroups of $G$ are small in the sense that any torsion subgroup of $G$ is artinian. In particular, if $\varphi$ is surjective and $G$ additionaly does not have non-trivial normal torsion subgroups, then $\varphi$ is continuous. As an application we obtain results concerning the continuity of group homomorphisms from locally compact Hausdorff groups to many groups from geometric group theory, in particular to automorphism groups of right-angled Artin groups and to Helly groups.