On preservation of automatic continuity

Abstract

A group G is called automatically continuous if any homomorphism from a completely metrizable or locally compact Hausdorff group to G has open kernel. In this paper, we study preservation of automatic continuity under group-theoretic constructions, focusing mainly on groups of size less than continuum. In particular, we consider group extensions and graph products. As a consequence, we establish automatic continuity of virtually poly-free groups, and hence of non-exceptional spherical Artin groups.On the other hand, we show that if G is automatically continuous, then so is any finitely generated residually G group, hence, for instance, all finitely generated residually free groups are automatically continuous.

Abstract group actions of locally compact groups on CAT(0) spaces

“…It is known that a rightangled Artin group A acts freely on the universal cover of the corresponding Salvetti complex S which is a finite dimensional CAT.0/ cube complex [13,Theorem 3.6]. Hence, as an immediate application of Proposition C we obtain a geometric proof for the following result, which can be proved by using Dudley's arguments in [18] and can be found in [16,Corollary 3.13].…”

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.…”

Inverse limit slender groups

“…They showed that many interesting groups, including free groups, free abelian groups and torsion-free word-hyperbolic groups, are lch-slender and cmslender and that for abelian groups we recover the usual slender groups. Several types and properties of automatically continuous were studied in [10,24]. Most recently, Corson and Varghese [11] strengthened the connection between slender groups and automatic continuity by showing that a group is lch-slender if it is torsion-free and it does not contain Q or p-adic integers for any p.…”