Automatic continuity for groups whose torsion subgroups are small

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