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