Abstract homomorphisms from locally compact groups to discrete groups

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.

Automatic continuity for groups whose torsion subgroups are small

“…We want to do this with geometric means. This is inspired by [KV19]. Given a graph product G Γ , there exists a finite dimensional right-angled building X Γ on which G Γ acts isometrically [Dav98, Theorem 5.1], see also [KV19, Section 3.5].…”

“…Given a finite simplicial graph Γ = (V, E) and a collection of groups G = {G u | u ∈ V }, the graph product G Γ is defined as the quotient ( * u∈V G u )/ [G v , G w ] for {v, w} ∈ E . Kramer and the third author proved in [KV19] that if the vertex set of Γ is not equal to S ∪ {w ∈ V | {v, w} ∈ E for all v ∈ S} where the subgraph generated by S is complete, then Epi(LCG, G Γ ) = cEpi(LCG, G Γ ). Further, the second and third author proved in [MV20] that if G is a subgroup of a CAT(0) group whose torsion subgroups are finite and G does not have non-trivial finite normal subgroups, then Epi(LCG, G) = cEpi(LCG, G) by geometric means.…”

Abstract homomorphisms from some topological groups to acylindrically hyperbolic groups

“…Free (abelian) groups were classically shown to be cm-slender, lcH-slender and n-slender (see [23] and [34]). More recent work has shown that torsion-free word hyperbolic groups, right-angled Artin groups, braid groups and many other groups satisfy various of these slenderness conditions (see [17,20,39,44]). Note that a group which is either n-slender, cm-slender, or lccH-slender must be torsion-free.…”

“…Another important result in this direction is a theorem of Nikolov and Segal [45, Theorem 1.1], which says that every abstract homomorphism from a finitely generated in topological sense profinite group to any profinite group is continuous. Papers [14,17,20,23,39,43,49] deal with automatic continuity of abstract homomorphisms from locally compact Hausdorff groups to some discrete groups; papers [17,20] also deal with completely metrizable groups as domains.…”

On parabolic subgroups of Artin groups

“…Our focus here is on continuity of group homomorphisms from locally compact Hausdorff groups into Artin groups. Many types of Artin groups are known to be lcH-slender such as right-angled Artin groups [KV19], [CK20], [MV20], Artin groups of spherical type [KV19] and more generally Artin groups of FC-type [KMV21]. We conjecture:…”