A note on automatic continuity

TL;DR

This paper presents new results in automatic continuity, showing that certain homomorphisms from metrizable groups to specific groups have open kernels, unifying various diagonalization concepts in the field.

Contribution

It introduces unified results on automatic continuity, extending previous diagonalization concepts to broader classes of groups and homomorphisms.

Findings

Homomorphisms from completely metrizable groups to Thompson's group F have open kernels.

Similar results hold for Baumslag-Solitar and torsion-free hyperbolic groups.

Unification of diagonalization concepts in automatic continuity theory.

Abstract

We present new results regarding automatic continuity, unifying some diagonalization concepts that have been developed over the years. For example, any homomorphism from a completely metrizable topological group to Thompson's group $F$ has open kernel. A similar claim holds when $F$ is replaced with a Baumslag-Solitar group or a torsion-free word hyperbolic group.