Recognizing the second derived subgroup of free groups

Conner, Gregory R.; Herfort, Wolfgang; Kent, Curtis; Pavešić, Petar

doi:10.1016/j.jalgebra.2018.09.024

Cited by 4 publications

(4 citation statements)

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“…One can consider intersections of homomorphisms to other groups as well, for example finite or torsion-free groups. In [11] we gave an intrinsic description of he second commutator subgroup F ′′ n using homomorphisms to the Baumslag-Solitar group B(1, n).…”

Section: Algebraic Applicationsmentioning

confidence: 99%

“…As before, we replace F ′′ with a suitable inverse limit group. We already mentioned that the second derived group of a free group F can be described as a kernel, F ′′ = Ker B (F ) (see [11,Theorem 1]) for any solvable, deficiency 1 group B that is not virtually abelian. By work of Wilson [31], any solvable deficiency 1 group is isomorphic to a Baumslag-Solitar group B(1, m) for some m. Since B(1, m) is nc-slender (see [5]), Lemma 4.1 implies that Ker B ( F ) = lim ← − Ker B (F n ) for every group B that is solvable, of deficiency 1 and is not virtually abelian.…”

Section: Algebraic Applicationsmentioning

confidence: 99%

“…Given groups G and A, we let Ker A (G) be the intersection of kernels of all homomorphisms from G to A. If G is a free group and B(1, n) is a Baumslag-Solitar group, then Ker Z (G) is exactly the commutator subgroup of G and Ker B(1,n) (G) is exactly the second derived subgroup of G, see [11]. However, F is locally free but non-free and the corresponding Ker Z ( F ) properly contains the commutator subgroup of F .…”

Section: Introductionmentioning

confidence: 99%

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Uncountable groups and the geometry of inverse limits of coverings

Conner¹,

Herfort²,

Kent³

et al. 2021

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In this paper we develop a new approach to the study of uncountable fundamental groups by using Hurewicz fibrations with the unique path-lifting property (lifting spaces for short) as a replacement for covering spaces. In particular, we consider the inverse limit of a sequence of covering spaces of X. It is known that the path-connectivity of the inverse limit can be expressed by means of the derived inverse limit functor lim ← − 1 , which is, however, notoriously difficult to compute when the π1(X) is uncountable. To circumvent this difficulty, we express the set of path-components of the inverse limit, X, in terms of the functors lim ← − and lim ← − 1 applied to sequences of countable groups arising from polyhedral approximations of X.A consequence of our computation is that path-connectedness of lifting space implies that π1( X) supplements π1(X) in π1(X) where π1(X) is the inverse limit of fundamental groups of polyhedral approximations of X. As an application we show that Gwhere F is the canonical inverse limit of finite rank free groups, G is the fundamental group of the Hawaiian Earring, and KerA( F ) is the intersection of kernels of homomorphisms from F to A.

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Section: Algebraic Applicationsmentioning

confidence: 99%

Section: Algebraic Applicationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Uncountable groups and the geometry of inverse limits of coverings

Conner¹,

Herfort²,

Kent³

et al. 2021

Preprint

Self Cite

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“…As mentioned before, part of our motivation for this work is of topological nature. In particular, our study of inverse limits of covering spaces naturally requires good understanding of homomorphisms from inverse limits of groups (see Conner, Herfort, Kent and Pavešić [5,6] for further information). In the final part of this paper we present two applications of our work on (shape) homotopy groups and (Čech) cohomology groups of Peano continua (see Section 6 for relevant definitions).…”

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confidence: 99%

Inverse limit slender groups

et al. 2023

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Classically, an abelian group G is said to be slender if every homomorphism from the countable product Z N N to G factors through the projection to some finite product Z n . Various authors have proposed generalizations to non-commutative groups; this has resulted 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 they are 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 coefficient theorem for Čech cohomology with coefficients in a slender group.

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