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Automatic continuity for groups whose torsion subgroups are small

We prove that a group homomorphism $φ\colon L\to G$ 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 $φ(N)$ is a torsion group provided that $G$ does not include $\mathbb{Q}$ or the $p$-adic integers $\mathbb{Z}_p$ or the Prüfer $p$-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 $φ$ is surjective and $G$ additionaly does not have non-trivial normal torsion subgroups, then $φ$ 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.