Automatic Continuity of Abstract Homomorphisms Between Locally Compact and Polish Groups

Braun, Oskar; Hofmann, Karl Heinrich; Krämer, Ludwig

doi:10.1007/s00031-019-09537-4

Cited by 3 publications

(2 citation statements)

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“…Another important result in this direction is a theorem of Nikolov and Segal [45, Theorem 1.1], which says that every abstract homomorphism from a finitely generated in topological sense profinite group to any profinite group is continuous. Papers [14,17,20,23,39,43,49] deal with automatic continuity of abstract homomorphisms from locally compact Hausdorff groups to some discrete groups; papers [17,20] also deal with completely metrizable groups as domains.…”

Section: Introductionmentioning

confidence: 99%

Abstract homomorphisms from some topological groups to acylindrically hyperbolic groups

Bogopolski

Corson

2021

Math. Ann.

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We describe homomorphisms ϕ : H → G for which G is acylindrically hyperbolic and H is a topological group which is either completely metrizable or locally countably compact Hausdorff. It is shown that, in a certain sense, either the image of ϕ is small or ϕ is almost continuous. We also describe homomorphisms from the Hawaiian earring group to G as above. We prove a more precise result for homomorphisms ϕ : H → Mod( ), where H is as above and Mod( ) is the mapping class group of a connected compact surface . In this case there exists an open normal subgroup V ≤ H such that ϕ(V ) is finite. We also prove the analogous statement for homomorphisms ϕ : H → Out(G), where G is a one-ended hyperbolic group. Some automatic continuity results for relatively hyperbolic groups and fundamental groups of graphs of groups are also deduced. As a by-product, we prove that the Hawaiian earring group is acylindrically hyperbolic, but does not admit any universal acylindrical action on a hyperbolic space.

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Section: Introductionmentioning

confidence: 99%

Abstract homomorphisms from some topological groups to acylindrically hyperbolic groups

Bogopolski

Corson

2021

Math. Ann.

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“…not necessarily continuous) homomorphism ϕ : L → G automatically continuous? There are many results in this direction in the literature, see [11], [17], [21], [32] or [35]. In particular, Dudley [21] proved that every abstract homomorphism from a locally compact group to a free group is automatically continuous.…”

Section: Introductionmentioning

confidence: 99%

Abstract homomorphisms from locally compact groups to discrete groups

Krämer

Varghese

2019

Journal of Algebra

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We show that every abstract homomorphism ϕ from a locally compact group L to a graph product G Γ , endowed with the discrete topology, is either continuous or ϕ(L) lies in a 'small' parabolic subgroup. In particular, every locally compact group topology on a graph product whose graph is not 'small' is discrete. This extends earlier work by Morris-Nickolas.We also show the following. If L is a locally compact group and if G is a discrete group which contains no infinite torsion group and no infinitely generated abelian group, then every abstract homomorphism ϕ : L → G is either continuous, or ϕ(L) is contained in the normalizer of a finite nontrivial subgroup of G. As an application we obtain results concerning the continuity of homomorphisms from locally compact groups to Artin and Coxeter groups.

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