On preservation of automatic continuity

TL;DR

This paper investigates the preservation of automatic continuity in groups under various constructions, establishing automatic continuity for certain classes like virtually poly-free and residually free groups, and analyzing the conditions for its inheritance.

Contribution

It demonstrates that automatic continuity is preserved under group extensions and graph products, and proves automatic continuity for classes such as virtually poly-free and residually free groups.

Findings

Automatic continuity is preserved under group extensions and graph products.

Virtually poly-free groups and non-exceptional spherical Artin groups are automatically continuous.

Finitely generated residually free groups are automatically continuous.

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.