Abstract homomorphisms from locally compact groups to discrete groups

2019/02/21 by Kramer, Linus, Varghese, Olga
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1902.07962

Abstract

We show that every abstract homomorphism φ from a locally compact group L to a graph product G_Γ, endowed with the discrete topology, is either continuous or φ(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 φ:L→ G is either continuous, or φ(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.

Related

• Automatic continuity for groups whose torsion subgroups are small
• On locally compact groups with zero
• Locally compact groups approximable by subgroups isomorphic to \mathbb Z or \mathbb R
• On small abstract quotients of Lie groups and locally compact groups
• Automatic continuity for homomorphisms into free products