Automatic continuity for groups whose torsion subgroups are small

Abstract

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 Z p 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.

Locally elliptic actions, torsion groups, and nonpositively curved spaces

“…in the theory of Polish spaces. It is a direct (but non-obvious) consequence of Theorem G and [KMV21].…”

“…(V) The question of automatic continuity for groups; see e.g. [KMV21]. Except the group theory the question has origins e.g.…”