Automatic continuity for groups whose torsion subgroups are small

Author:

Keppeler Daniel1,Möller Philip1,Varghese Olga2

Affiliation:

1. Department of Mathematics , University of Münster , Einsteinstraße 62, 48149 Münster , Germany

2. Department of Mathematics , Otto-von-Guericke University of Magdeburg , Universitätsplatz 2, 39106 Magdeburg , Germany

Abstract

Abstract We prove that a group homomorphism φ : L G \varphi\colon L\to G from a locally compact Hausdorff group 𝐿 into a discrete group 𝐺 either is continuous, or there exists a normal open subgroup N L N\subseteq L such that φ ( N ) \varphi(N) is a torsion group provided that 𝐺 does not include ℚ or the 𝑝-adic integers Z p \mathbb{Z}_{p} or the Prüfer 𝑝-group Z ( p ) \mathbb{Z}(p^{\infty}) for any prime 𝑝 as a subgroup, and if the torsion subgroups of 𝐺 are small in the sense that any torsion subgroup of 𝐺 is artinian. In particular, if 𝜑 is surjective and 𝐺 additionally does not have non-trivial normal torsion subgroups, then 𝜑 is continuous. As an application, we obtain results concerning the continuity of group homomorphisms from locally compact Hausdorff groups to many groups from geometric group theory, in particular to automorphism groups of right-angled Artin groups and to Helly groups.

Publisher

Walter de Gruyter GmbH

Subject

Algebra and Number Theory

Reference48 articles.

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