Vol. 289, No. 1, 2017

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ISSN: 0030-8730
Molino theory for matchbox manifolds

Jessica Dyer, Steven Hurder and Olga Lukina

Vol. 289 (2017), No. 1, 91–151
DOI: 10.2140/pjm.2017.289.91
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. We develop the Molino theory for all equicontinuous matchbox manifolds. Our work extends the Molino theory developed by Álvarez López and Moreira Galicia, which required the hypothesis that the holonomy actions for these spaces satisfy the strong quasianalyticity condition. The methods of this paper are based on the authors’ previous work 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 quasianalytic 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 nontrivial Molino sequences, and other interesting properties.

Keywords
equicontinuous foliations, Molino theory, minimal Cantor actions, group chains, profinite groups
Mathematical Subject Classification 2010
Primary: 20E18, 37B45, 57R30
Secondary: 37B05, 58H05
Milestones
Received: 17 October 2016
Revised: 23 December 2016
Accepted: 23 December 2016
Published: 12 May 2017
Authors
Jessica Dyer
Department of Mathematics
Tufts University, School of Arts and Sciences
503 Boston Avenue
Bromfield-Pearson
Medford, MA 02155
United States
Steven Hurder
Department of Mathematics, Statistics and Computer Science
University of Illinois at Chicago
322 SEO (m/c 249)
851 S. Morgan Street
Chicago, IL 60607-7045
United States
Olga Lukina
Department of Mathematics, Statistics and Computer Science
University of Illinois at Chicago
322 SEO (m/c 249)
851 S. Morgan Street
Chicago, IL 60607-7045
United States