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Mathematics 2016, 4(1), 18; doi:10.3390/math4010018

Dynamics and the Cohomology of Measured Laminations

Instituto de Matemática, Universidade Federal do Rio de Janeiro, Rio de Janeiro 21941-909, Brazil
Academic Editor: Yuli B. Rudyak
Received: 28 November 2015 / Revised: 19 February 2016 / Accepted: 4 March 2016 / Published: 15 March 2016
(This article belongs to the Special Issue Algebraic and Geometric Topology)
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Abstract

In this paper, the interconnection between the cohomology of measured group actions and the cohomology of measured laminations is explored, the latter being a generalization of the former for the case of discrete group actions and cocycles evaluated on abelian groups. This relation gives a rich interplay between these concepts. Several results can be adapted to this setting—for instance, Zimmer’s reduction of the coefficient group of bounded cocycles or Fustenberg’s cohomological obstruction for extending the ergodicity \(\mathbb{Z}\)-action to a skew product relative to an \(S^{1}\) evaluated cocycle. Another way to think about foliated cocycles is also shown, and a particular application is the characterization of the existence of certain classes of invariant measures for smooth foliations in terms of the \(L^{\infty}\)-cohomology class of the infinitesimal holonomy. View Full-Text
Keywords: foliations; cohomology; group action; foliated cocycles; invariant measures foliations; cohomology; group action; foliated cocycles; invariant measures
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Meniño Cotón, C. Dynamics and the Cohomology of Measured Laminations. Mathematics 2016, 4, 18.

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