Solenoidal manifolds (Q2879073)

In the paper under review, the author discusses the geometry of solenoidal surfaces (or more generally, laminar surfaces) and corresponding Teichmüller spaces. A \(k\)-dimensional laminar manifold is a compact Hausdorff space which is locally homeomorphic to a \(k\)-disk times a compact totally disconnected space. A solenoidal manifold is a laminar manifold which is locally homeomorphic to a disk times the Cantor set. We can speak of the structures on a laminar manifold, such as smooth, Riemannian, complex etc.NEWLINENEWLINE\noindent It is shown, rather sketchily, that any oriented solenoidal \(1\)-manifolds is the boundary of an oriented solenoidal surface (Theorem \(1\)). As for laminar (or solenoidal) surfaces, several theorems mostly with sketches of the proof are stated. Theorem \(3\) states \textit{A. Candel}'s result [Ann. Sci. Éc. Norm. Supér. (4) 26, No. 4, 489--516 (1993; Zbl 0785.57009)]: for any Riemannian laminar surface, the universal cover of every leaf is conformally the disk, or there is a nontrivial transverse invariant measure (both can happen). Also due to Candel (1993) and \textit{A. Verjovsky} [Contemp. Math. 58, 233--253 (1987; Zbl 0619.32017)], if every leaf of a Riemannian laminar surface is conformally covered by the disk, then the unique hyperbolic metric on each leaf is transversally continuous (Theorem \(4\)). The author's result [in: Topological methods in modern mathematics. Proceedings of a symposium in honor of John Milnor's sixtieth birthday, held at the State University of New York at Stony Brook, USA, June 14-June 21, 1991. Houston, TX: Publish or Perish, Inc.. 543--564 (1993; Zbl 0803.58018)] states that the space of hyperbolic structures on a laminar surface up to isometries isotopic to the identity (``Teichmüller space'') has the structure of a separable complex Banach manifold. The metric is the natural Teichmüller metric based on the minimal conformal distortion of a map, and the isotopy classes of homeomorphisms preserving a chosen leaf act by isometries of this Teichmüller space and play the role of the Teichmüller modular group in the classical case (Theorem \(5\)). As a corollary, it is noted that there is an analog of Riemann's moduli space if and only if this Teichmüller modular group acts appropriately discontinuous. Also the author considers the universal hyperbolic solenoid, i.e., the inverse limit solenoidal surface \(S\) of the inverse system of finite covers over a compact surface of genus \(> 1\). In this case the Teichmüller space for \(S\) is non-Hausdorff and its Hausdorff quotient is a point. This result is based on Kahn-Marković's affirmation of the Ehrenpreis Conjecture.NEWLINENEWLINE\noindent It should be noted that several remarks and comments on this paper have been published by \textit{A. Verjovsky} [J. Singul. 9, 245--251 (2014; Zbl 1330.57043)].