Abstract
Riemannian foliations are characterized as those foliations whose holonomy pseudogroup consists of local isometries of a Riemannian manifold. Their main structural features are well understood since the work of Molina. In this paper we analyze the more general concept of equicontinuous pseudogroup of homeomorphisms, which gives rise to the notion of equicontinuous foliated space. We show that equicontinuous foliated spaces have structural properties similar to those known for Riemannian foliations: the universal covers of their leaves are in the same quasi-isometry class, leaf closures are homogeneous spaces, and the holonomy pseudogroup is indeed given by local isometries.
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Alcalde Cuesta F.: Groupoï de d’homotopie d’un feuilletage riemannien et réalisation symplectique de certaines variétés de Poisson. Publ. Mat. 33, 395–410 (1989)
Álvarez López, J.A., Candel, A.: Topological description of Riemannian foliations with dense leaves (preprint)
Álvarez López, J.A., Candel, A.: Generic geometry of leaves (Forthcomming preprint)
Block J., Weinberger S.: Aperiodic tilings, positive scalar curvature, and amenability of spaces. J. Am. Math. Soc. 5, 907–918 (1992)
Candel, A., Conlon, L.: Foliations I, Graduate Studies in Mathematics. American Mathematical Society, Providence (2000)
Goodman S.E., Plante J.F.: Holonomy and averaging in foliated sets. J. Diff. Geom. 14, 401–407 (1979)
Gromov M.: Asymptotic invariants of infinite groups. In: Niblo, G.A., Roller, M.A. (eds) Geometric Group Theory vol. 2, Cambridge University Press, Cambridge (1993)
Haefliger, A.: Pseudogroups of local isometries. In: Cordero, L.A. (ed.) Differential Geometry (Santiago de Compostela 1984). Research Notes in Math., vol. 131, pp. 174–197. Pitman Advanced Pub. Program, Boston (1985)
Haefliger A. : Leaf closures in Riemannian foliations. In: Matsumoto, Y., Mizutani, T., Morita S., S. (eds) A Fête on Topology, pp. 3–32. Academic Press, New York (1988)
Haefliger, A.: Foliations and compactly generated pseudogroups (2001, preprint)
Hector, G., Hirsch, U.: Introduction to the geometry of foliations, Part A. In: Aspects of Mathematics, vol. E1. Friedr. Vieweg and Sohn, Braunschweig (1981)
Hector, G., Hirsch, U.: Introduction to the geometry of foliations, Part B. In: Aspects of Mathematics, vol. E3. Friedr. Vieweg and Sohn, Braunschweig (1983)
Hirsch M.: Differential Topology, Graduate Texts in Mathematics, vol. 33. Springer, New York (1976)
Hurder S.: Coarse geometry of foliations. In: Mizutani, T., Masuda, K., Matsumoto, S., Inaba, T., Tsuboi, T., Mitsumatsu, Y. (eds) Geometric Study of Foliations (Tokyo 1993), pp. 35–96. World Scientific Publishing Co. Pte. Ltd, Singapore (1994)
Hurder S., Katok A.: Ergodic theory and Weil measures for foliations. Ann. Math. 126, 221–275 (1987)
Kanai M.: Rough isometries, and combinatorial approximations of geometries of non-compact manifolds. J. Math. Soc. Jpn 37, 391–413 (1985)
Kellum M.: Uniformly quasi-isometric foliations. Ergodic Theory Dyn. Syst. 13, 101–122 (1993)
Molino, P.: Riemannian Foliations (with appendices by Cairns, G., Carrière, Y., Ghys, E., Salem, E., Sergiescu, V.). Progress in Mathematics, vol. 73. Birkhäuser, Boston (1988)
Moore C.C., Schochet C.: Global Analysis on Foliated Spaces, MSRI Publications, vol. 9. Springer, New York (1988)
Munkres J.R.: Topology: a First Course. Prentice-Hall, Inc., Englewood Cliffs (1975)
Nagata J.: Modern General Topology, 2nd revised edn. Noth-Holland Publishing Company, Amsterdam (1974)
Robinson A.: Non-standard Analysis, (Reprint of the 1974 Edition). Princeton University Press, Princeton (1996)
Plante J.F.: Foliations with measure preserving holonomy. Ann. Math. 102, 327–361 (1975)
Roe J.: Coarse cohomology and index theory on complete riemannian manifolds. Mem. Amer. Math. Soc. 104(497), x+90 (1993)
Sacksteder R.: Foliations and pseudogroups. Am. J. Math. 87, 79–102 (1965)
Smirnov Y.M.: On metrization of topological spaces. Am. Math. Soc. Transl. Ser. One 8, 62–77 (1953)
Steen L.A., Seebach J.A. Jr: Counterexamples in Topology, 2nd edn. Springer, New York (1978)
Tarquini C.: Feuilletages conformes. Ann. Inst. Fourier 54, 453–480 (2004)
Veech W.: Topological dynamics. Bull. Am. Math. Soc. 83, 775–830 (1977)
Weil, A.: L’Integration dans les Groupes Topologiques et ses Applications. Actualités Scientifiques et Industrielles, no. 1145, 2nd edn. Publications de l’Institut de Mathematique de l’Universite de Strasbourg 4, Hermann, Paris (1951)
Willard S.: General Topology. Addison-Wesley Publishing Co., Reading (1970)
Winkelnkemper H.E.: The graph of a foliation. Ann. Global Anal. Geom. 1, 51–75 (1983)
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Research of J. A. Álvarez López was supported by DGICYT Grant PB95-0850. Research of A. Candel was supported by NSF Grants.
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López, J.A.Á., Candel, A. Equicontinuous foliated spaces. Math. Z. 263, 725–774 (2009). https://doi.org/10.1007/s00209-008-0432-4
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DOI: https://doi.org/10.1007/s00209-008-0432-4