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.2010 Mathematics Subject Classification. Primary 03E75, 54H11; Secondary 20E06.
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.
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