The discriminant invariant of Cantor group actions

Dyer, Jessica; Hurder, Steve; Lukina, Olga

doi:10.48550/arxiv.1509.06227

Cited by 3 publications

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“…The action (H U , U, Φ U ) obtained in Proposition 2.14 is an equicontinuous minimal Cantor system in the terminology of [21]. COROLLARY 2.16.…”

Section: Given a Clopen Setmentioning

confidence: 98%

“…It is possible that the restricted holonomy pseudo⋆group of a weak solenoid is SQA for a suitable choice of a section, though the global holonomy action of the solenoid is not SQA. This is true, for example, in the case of [21,Example 7.6]. Thus, the global holonomy action of a weak solenoid may be LSQA, though need not satisfy the SQA condition for a given section X.…”

Section: Definition 33mentioning

confidence: 99%

“…The asymptotic discriminant of Φ is the the tail equivalence class, I(Φ) = [I(U, x)] ∞ , for a choice of the basepoint x ∈ X and a choice of an adapted neighborhood basis U at x. The explanation for the notation "asymptotic discriminant" will be made clear in Sections 6.2 and 6.3, where we give an interpretation of this class in terms of the group chain model for the action and the notion of the discriminant group for such actions, as introduced in the works [20,21,23].…”

Section: Proof By Lemma 24 There Existsmentioning

confidence: 99%

“…A foliated space is a natural generalization of the notion of a smooth foliation F of a compact connected manifold M . A foliated space M for which the local transversals to the foliation F M are totally disconnected is called a matchbox manifold, and the leaves of F M are the path connected components of M. Matchbox manifolds were introduced in the 1-dimensional case in the works [2,3,4], have been studied in the higher dimensional cases in the works [15,16,17,18,19,21,22,23].…”

Section: Introductionmentioning

confidence: 99%

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Wild solenoids

Hurder¹,

Lukina

2018

Trans. Amer. Math. Soc.

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Dedicated to the memory of James T. Rogers, Jr.Abstract. A weak solenoid is a foliated space defined as the inverse limit of finite coverings of a closed compact manifold M . The monodromy of a weak solenoid defines an equicontinuous minimal action on a Cantor space X by the fundamental group G of M . The discriminant group of this action is an obstruction to this action being homogeneous. The discriminant vanishes if the group G is abelian, but there are examples of actions of nilpotent groups for which the discriminant is non-trivial. The action is said to be stable if the discriminant group remains unchanged for the induced action on sufficiently small clopen neighborhoods in X. If the discriminant group never stabilizes as the diameter of the clopen set U tends to zero, then we say that the action is unstable, and the weak solenoid which defines it is said to be wild. In this work, we show two main results in the course of our study of the properties of the discriminant group for Cantor actions. First, the tail equivalence class of the sequence of discriminant groups obtained for the restricted action on a neighborhood basis system of a point in X defines an invariant of the return equivalence class of the action, called the asymptotic discriminant, which is consequently an invariant of the homeomorphism class of the weak solenoid. Second, we construct uncountable collections of wild solenoids with pairwise distinct asymptotic discriminant invariants for a fixed base manifold M , and hence fixed finitely-presented group G, which are thus pairwise non-homeomorphic. The study in this work is the continuation of the seminal works on homeomorphisms of weak solenoids by Rogers and Tollefson in 1971, and is dedicated to the memory of Jim Rogers.

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“…The action (H U , U, Φ U ) obtained in Proposition 2.14 is an equicontinuous minimal Cantor system in the terminology of [21]. COROLLARY 2.16.…”

Section: Given a Clopen Setmentioning

confidence: 98%

Section: Definition 33mentioning

confidence: 99%

Section: Proof By Lemma 24 There Existsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Wild solenoids

Hurder¹,

Lukina

2018

Trans. Amer. Math. Soc.

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“…The hypothesis that a matchbox manifold M is manifold-like does not imply that M is homogeneous, as the "discriminant obstruction" to homogeneity for a weak solenoid is supported in arbitrarily small open neighborhoods of points in M . The discriminant invariant was introduced and studied in the works in [15,16,17,24].…”

Section: Introductionmentioning

confidence: 99%

Manifold-like matchbox manifolds

Clark

Hurder

Lukina

2019

Proc. Amer. Math. Soc.

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A matchbox manifold is a generalized lamination, which is a continuum whose arccomponents define the leaves of a foliation of the space. The main result of this paper implies that a matchbox manifold which is manifold-like must be homeomorphic to a weak solenoid. DEFINITION 1.2. A continuum X is said to be manifold-like, if there exists n ≥ 1 such that for every ǫ > 0, there exists M ǫ ∈ M(n) and an ǫ-map f ǫ : X → M ǫ .The study of the properties of ǫ-maps and manifold-like continua has a long history in the study of the topology of spaces. Eilenberg showed in [19] that an ǫ-map, for ǫ > 0 sufficiently small, admits a left approximate inverse. Ganea studied the properties of compact, locally connected manifoldlike ANR's of dimension n in [20], and showed that such a space has the homotopy type of a closed n-manifold. Deleanu [12,13] showed that an n-dimensional connected polyhedron which is manifoldlike is a closed pseudo-manifold. Bob Edwards gave in his 1978 ICM address [18] an overview of the further applications of ǫ-approximations and homeomorphisms.Mardešić and Segal [26,27] studied the properties of manifold-like connected polyhedron, and gave conditions under which such spaces must be a topological manifold. These authors used a technique of approximation of the given continuum by an inverse limit of spaces, and noted that their results do not apply to a continuum which is not locally connected, such as the dyadic solenoid.The goal of this work is to characterize a class of manifold-like continua for which the Mardešić and Segal results do not apply. These are the matchbox manifolds as studied by the authors in [9,10,11], and discussed below. Our study of matchbox manifolds in these works was inspired by a result of Bing in [6]. Recall that a topological space X is homogeneous if for every x, y ∈ X, there exists a homeomorphism h : X → X such that h(x) = y. THEOREM 1.3. Let X be a homogeneous, circle-like continuum that contains an arc. Then either X is homeomorphic to the circle S 1 , or to an inverse limit of coverings of S 1 .

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Molino theory for matchbox manifolds

2017

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Abstract. A matchbox manifold is a foliated space with totally disconnected transversals, and an equicontinuous matchbox manifold is the generalization of Riemannian foliations for smooth manifolds in this context. In this paper, we develop the Molino theory for all equicontinuous matchbox manifolds. Our work extends the Molino theory developed in the work ofÁlvarez López and Moreira Galicia which required the hypothesis that the holonomy actions for these spaces satisfy the strong quasi-analyticity condition. The methods of this paper are based on the authors' previous works on the structure of weak solenoids, and provide many new properties of the Molino theory for the case of totally disconnected transversals, and examples to illustrate these properties. In particular, we show that the Molino space need not be uniquely well-defined, unless the global holonomy dynamical system is stable, a notion defined in this work. We show that examples in the literature for the theory of weak solenoids provide examples for which the strong quasi-analytic condition fails. Of particular interest is a new class of examples of equicontinuous minimal Cantor actions by finitely generated groups, whose construction relies on a result of Lubotzky. These examples have non-trivial Molino sequences, and other interesting properties.

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The discriminant invariant of Cantor group actions

Cited by 3 publications

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Wild solenoids

Wild solenoids

Manifold-like matchbox manifolds

Molino theory for matchbox manifolds

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