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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Homogeneous matchbox manifolds


Authors: Alex Clark and Steven Hurder
Journal: Trans. Amer. Math. Soc. 365 (2013), 3151-3191
MSC (2010): Primary 57S10, 54F15, 37B10, 37B45; Secondary 57R05
Published electronically: October 3, 2012
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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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Additional Information

Alex Clark
Affiliation: Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, United Kingdom
Email: adc20@le.ac.uk

Steven Hurder
Affiliation: Department of Mathematics, University of Illinois at Chicago, 322 SEO (m/c 249), 851 S. Morgan Street, Chicago, Illinois 60607-7045
Email: hurder@uic.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05753-9
PII: S 0002-9947(2012)05753-9
Keywords: Solenoids, matchbox manifold, laminations, equicontinuous foliation, Effros Theorem, foliations
Received by editor(s): June 28, 2010
Received by editor(s) in revised form: June 8, 2011, July 12, 2011, and October 24, 2011
Published electronically: October 3, 2012
Additional Notes: Both authors were supported by NWO travel grant 040.11.132
The first author was supported in part by EPSRC grant EP/G006377/1
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.