A Nunke type classification in the locally compact setting

Corson, Samuel M.; Varghese, Olga

doi:10.1016/j.jalgebra.2020.07.011

Cited by 5 publications

(8 citation statements)

References 8 publications

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“…We will always assume that topological groups have the Hausdorff property. There are several results in this direction in the literature, see [5,15,17,18,23,27]. Here, the group G will be a locally compact group while H will be the isometry group of a CAT.0/ space equipped with the discrete topology.…”

Section: Introductionmentioning

confidence: 99%

“…Using the fact that an abstract group homomorphism ' W L ! H is continuous if and only if H is torsion-free and does not include Q or the p-adic integers Z p for any prime p (see [17]), the result in [23] was recently generalized in [34] to a larger class consisting of all groups G with the following properties: torsion subgroups of G are finite, and Q and the p-adic integers Z p for any prime p are not a subgroup of G.…”

Section: Introductionmentioning

confidence: 99%

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Abstract group actions of locally compact groups on CAT(0) spaces

Möller

Varghese

2022

Groups Geom. Dyn.

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We study abstract group actions of locally compact Hausdorff groups on CAT.0/ spaces. Under mild assumptions on the action we show that it is continuous or has a global fixed point. This mirrors results by Dudley and Morris-Nickolas for actions on trees. As a consequence we obtain a geometric proof for the fact that any abstract group homomorphism from a locally compact Hausdorff group into a torsion-free CAT.0/ group is continuous.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Abstract group actions of locally compact groups on CAT(0) spaces

Möller

Varghese

2022

Groups Geom. Dyn.

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“…Several types and properties of automatically continuous were studied in [10,24]. Most recently, Corson and Varghese [11] strengthened the connection between slender groups and automatic continuity by showing that a group is lch-slender if it is torsion-free and it does not contain Q or p-adic integers for any p.…”

Section: Introductionmentioning

confidence: 99%

“…Since the class of inverse limit slender groups is intermediate, they must all coincide. On the other hand, the group Z N is lch-slender by [11] but is not slender. Corollary 3.11.…”

mentioning

confidence: 97%

“…In a recent paper Corson and Varghese [11] proved that a group G is lch-slender if, and only if, G is torsion-free and it does not contain subgroups isomorphic to Q or to p-adic integers for any prime p. This coincides with the characterization of stout groups in Göbel [22,Theorem 4.1]. Following Göbel [22] a group G is stout if Homp ś i H i , Gq is trivial whenever tH i | i P Nu is a set of groups with trivial HompH i , Gq.…”

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confidence: 97%

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Inverse limit slender groups

Conner¹,

Herfort²,

Kent³

et al. 2021

Preprint

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Classically, an abelian group G is said to be slender if every homomorphism from the countable product Z N to G factors through the projection to some finite product Z n . Various authors have proposed generalizations to non-commutative groups, resulting in a plethora of similar but not completely equivalent concepts. In the first part of this work we present a unified treatment of these concepts and examine how are they related. In the second part of the paper we study slender groups in the context of co-small objects in certain categories, and give several new applications including the proof that certain homology groups of Barratt-Milnor spaces are cotorsion groups and a universal coefficients theorem for Čech cohomology with coefficients in a slender group.

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Automatic continuity for groups whose torsion subgroups are small

Keppeler

Möller

Varghese

2022

Journal of Group Theory

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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.

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A Nunke type classification in the locally compact setting

Cited by 5 publications

References 8 publications

Abstract group actions of locally compact groups on CAT(0) spaces

Abstract group actions of locally compact groups on CAT(0) spaces

Inverse limit slender groups

Automatic continuity for groups whose torsion subgroups are small

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