Invariant distributions and cohomology for geodesic flows and higher cohomology of higher-rank Anosov actions

Felipe Alberto Ramirez

doi:10.1016/j.jfa.2013.05.010

Invariant distributions and cohomology for geodesic flows and higher cohomology of higher-rank Anosov actions

Felipe Alberto Ramirez

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We are motivated by a conjecture of A. and S. Katok to study the smooth cohomologies of a family of Weyl chamber flows. The conjecture is a natural generalization of the Livšic Theorem to Anosov actions by higher-rank abelian groups; it involves a description of top-degree cohomology and a vanishing statement for lower degrees. Our main result, proved in Part II, verifies the conjecture in lower degrees for our systems, and steps in the “correct” direction in top degree. In Part I we study our “base case”: geodesic flows of finite volume hyperbolic manifolds. We describe obstructions (invariant distributions) to solving the coboundary equation in unitary representations of the group of orientation-preserving isometries of hyperbolic N-space, and we study Sobolev regularity of solutions. (One byproduct is a smooth Livšic Theorem for geodesic flows of hyperbolic manifolds with cusps.) Part I provides the tools needed in Part II for the main theorem.
© 2013 Elsevier Inc. All rights reserved.

Original language	English
Pages (from-to)	1002-1063
Journal	Journal of Functional Analysis
Volume	265
Issue number	6
Early online date	28 May 2013
DOIs	https://doi.org/10.1016/j.jfa.2013.05.010
Publication status	Published - 15 Sept 2013

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10.1016/j.jfa.2013.05.010

http://www.sciencedirect.com/science/article/pii/S0022123613001559

Cite this

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title = "Invariant distributions and cohomology for geodesic flows and higher cohomology of higher-rank Anosov actions",

abstract = "We are motivated by a conjecture of A. and S. Katok to study the smooth cohomologies of a family of Weyl chamber flows. The conjecture is a natural generalization of the Liv{\v s}ic Theorem to Anosov actions by higher-rank abelian groups; it involves a description of top-degree cohomology and a vanishing statement for lower degrees. Our main result, proved in Part II, verifies the conjecture in lower degrees for our systems, and steps in the “correct” direction in top degree. In Part I we study our “base case”: geodesic flows of finite volume hyperbolic manifolds. We describe obstructions (invariant distributions) to solving the coboundary equation in unitary representations of the group of orientation-preserving isometries of hyperbolic N-space, and we study Sobolev regularity of solutions. (One byproduct is a smooth Liv{\v s}ic Theorem for geodesic flows of hyperbolic manifolds with cusps.) Part I provides the tools needed in Part II for the main theorem.{\textcopyright} 2013 Elsevier Inc. All rights reserved.",

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AB - We are motivated by a conjecture of A. and S. Katok to study the smooth cohomologies of a family of Weyl chamber flows. The conjecture is a natural generalization of the Livšic Theorem to Anosov actions by higher-rank abelian groups; it involves a description of top-degree cohomology and a vanishing statement for lower degrees. Our main result, proved in Part II, verifies the conjecture in lower degrees for our systems, and steps in the “correct” direction in top degree. In Part I we study our “base case”: geodesic flows of finite volume hyperbolic manifolds. We describe obstructions (invariant distributions) to solving the coboundary equation in unitary representations of the group of orientation-preserving isometries of hyperbolic N-space, and we study Sobolev regularity of solutions. (One byproduct is a smooth Livšic Theorem for geodesic flows of hyperbolic manifolds with cusps.) Part I provides the tools needed in Part II for the main theorem.© 2013 Elsevier Inc. All rights reserved.

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DO - 10.1016/j.jfa.2013.05.010

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