The preprint "Automatic continuity for groups whose torsion subgroups are small" by Daniel Keppeler, Philip Möller and Olga Varghese has been recently uploaded to the arXiv!
Link to the paper: arXiv:2106.12547
We prove that a group homomorphism φ:L→G from a locally compact Hausdorff group L into a discrete group G either is continuous, or there exists a normal open subgroup N⊆L such that φ(N) is a torsion group provided that G does not include Q or the p-adic integers Zp or the Prüfer p-group Z(p∞) 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.