Homogeneous matchbox manifolds

Clark, Alex; Hurder, Steven

doi:10.1090/S0002-9947-2012-05753-9

Homogeneous matchbox manifolds
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by Alex Clark and Steven Hurder PDF

Trans. Amer. Math. Soc. 365 (2013), 3151-3191 Request permission

Abstract:

We prove that a homogeneous matchbox manifold is homeomorphic to a McCord solenoid, thereby proving a strong version of a conjecture of Fokkink and Oversteegen, which is a general form of a conjecture of Bing. A key step in the proof shows that if the foliation of a matchbox manifold has equicontinuous dynamics, then it is minimal. Moreover, we then show that a matchbox manifold with equicontinuous dynamics is homeomorphic to a weak solenoid. A result of Effros is used to conclude that a homogeneous matchbox manifold has equicontinuous dynamics, and the main theorem is a consequence. The proofs of these results combine techniques from the theory of foliations and pseudogroups, along with methods from topological dynamics and coding theory for pseudogroup actions. These techniques and results provide a framework for the study of matchbox manifolds in general, and exceptional minimal sets of smooth foliations.

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