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.
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Communicated by A. Constantin.
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The authors are supported by ERC Grant PCG-336983, Basque Government Grant IT974-16 and Spanish Government Grant MTM2017-86802-P.
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Corson, S.M., Kazachkov, I. On preservation of automatic continuity. Monatsh Math 191, 37–52 (2020). https://doi.org/10.1007/s00605-019-01281-x
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DOI: https://doi.org/10.1007/s00605-019-01281-x