# Abstract homomorphisms from locally compact groups to discrete groups

## TL;DR

This paper investigates the nature of abstract homomorphisms from locally compact groups to discrete groups, showing conditions under which such homomorphisms are continuous or have images in small subgroups, extending previous results.

## Contribution

It extends earlier work by characterizing when homomorphisms from locally compact groups to certain discrete groups are continuous or have small images, especially for graph products and specific classes of groups.

## Key findings

- Homomorphisms to graph products are either continuous or have images in small parabolic subgroups.
- Homomorphisms to groups with no infinite torsion or infinitely generated abelian subgroups are either continuous or have images in normalizers of finite subgroups.
- Results apply to homomorphisms into Artin and Coxeter groups, ensuring continuity under certain conditions.

## Abstract

We show that every abstract homomorphism $\varphi$ from a locally compact group $L$ to a graph product $G_\Gamma$, endowed with the discrete topology, is either continuous or $\varphi(L)$ 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 $\varphi:L\to G$ is either continuous, or $\varphi(L)$ 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.