Automatic continuity for groups whose torsion subgroups are small

2021/06/23 by Daniel Keppeler, Keppeler, Daniel, Philip Möller +3
Mathematics · #Advanced Topology and Set Theory #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Primary: 22D05 #Secondary: 20F65

paper · pdf · doi:10.48550/arxiv.2106.12547

openalex publication_date 2021/06/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that a group homomorphism φ\colon L→ 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 such that φ(N) is a torsion group provided that G does not include ℚ or the p-adic integers ℤ_p or the Prüfer p-group ℤ(p^∞) 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 additionaly 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.

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