We prove that a group homomorphism φ : L → G \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 ⊆ L N\subseteq L such that φ ⁢ ( N ) \varphi(N) is a torsion group provided that 𝐺 does not include ℚ or the 𝑝-adic integers Z p \mathbb{Z}_{p} or the Prüfer 𝑝-group Z ⁢ ( p ∞ ) \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 additionally 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.

We study abstract group actions of locally compact Hausdorff groups on CAT.0/ spaces. Under mild assumptions on the action we show that it is continuous or has a global fixed point. This mirrors results by Dudley and Morris-Nickolas for actions on trees. As a consequence we obtain a geometric proof for the fact that any abstract group homomorphism from a locally compact Hausdorff group into a torsion-free CAT.0/ group is continuous.

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