Abstract homomorphisms from locally compact groups to discrete groups

We show that every abstract homomorphism phi from a locally compact group L to a graph product G(Gamma), endowed with the discrete topology, is either continuous or phi(L) lies in a 'small' parabolic subgroup. In particular, every locally compact group topology on a graph product whose graph is not 'small' is discrete. This extends earlier work by Morris-Nickolas. We also show the following. If L is a locally compact group and if G is a discrete group which contains no infinite torsion group and no infinitely generated abelian group, then every abstract homomorphism phi : L -> G is either continuous, or phi(L) is contained in the normalizer of a finite nontrivial subgroup of G. As an application we obtain results concerning the continuity of homomorphisms from locally compact groups to Artin and Coxeter groups. (C) 2019 Elsevier Inc. All rights reserved.