Abstract
Matchbox manifolds are foliated spaces with totally disconnected transversals. Two matchbox manifolds which are homeomorphic have return equivalent dynamics, so that invariants of return equivalence can be applied to distinguish non-homeomorphic matchbox manifolds. In this work we study the problem of showing the converse implication: when does return equivalence imply homeomorphism? For the class of weak solenoidal matchbox manifolds, we show that if the base manifolds satisfy a strong form of the Borel Conjecture, then return equivalence for the dynamics of their foliations implies the total spaces are homeomorphic. In particular, we show that two equicontinuous $\mT^n$--like matchbox manifolds of the same dimension are homeomorphic if and only if their corresponding restricted pseudogroups are return equivalent. At the same time, we show that these results cannot be extended to include the "\emph{adic}-surfaces", which are a class of weak solenoids fibering over a closed surface of genus 2.
Highlights
A matchbox manifold is a compact, connected metrizable space M, equipped with a decomposition into leaves of constant dimension, so that the pair (M, F ) is a foliated space as defined in [9, 36], for which the local transversals to the foliation are totally disconnected
The dynamical and topological properties of matchbox manifolds have been studied in a series of works by the authors [11, 13, 14]
Matchbox manifolds arise naturally as exceptional minimal sets for foliations of compact manifolds, for example see [29, 30]; as the tiling spaces associated to repetitive, aperiodic tilings of Euclidean space Rn which have finite local complexity, for example see [3, 41, 42]; and they appear naturally in the study of group representation theory and index theory for leafwise elliptic operators for foliations, as discussed in the books [9, 36]
Summary
A matchbox manifold is a compact, connected metrizable space M, equipped with a decomposition into leaves of constant dimension, so that the pair (M, F ) is a foliated space as defined in [9, 36], for which the local transversals to the foliation are totally disconnected. This conclusion yields a connection between return equivalence for the foliations of SP and SQ and the homotopy types of the approximating manifolds in the presentations P and Q This requirement need not be imposed for the case of Y = Tn in Theorem 1.3, due to the algebraic properties of Zn. We note that the injectivity of the global holonomy maps implies that the fundamental groups π1(M0, x0) and π1(N0, y0) are residually finite. A homeomorphism between matchbox manifolds induces a quasi-isometry between the leaves of the respective foliations, equipped with the induced metrics It is a classical result of Plante [37] that the quasi-isometry class of a leaf is determined by its intersection with any transversal, and provides a general invariant of asymptotic return equivalence. The constructions of examples of wild solenoids in [31, Section 9] were shown to yield uncountable collections of non-homeomorphic weak solenoids, all with the same compact base manifold whose fundamental group is a higher rank lattice, and in particular is highly non-abelian
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