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Amenability, Kazhdan’s property and percolation for trees, groups and equivalence relations

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Abstract

We prove amenability for a broad class of equivalence relations which have trees associated to the equivalence classes. The proof makes crucial use of percolation on trees. We also discuss related concepts and results, including amenability of automorphism groups. A second main result is that no discrete subgroup of the automorphism group of a tree is isomorphic to the fundamental group of any closed manifoldM admitting a nontrivial connection-preserving, volume-preserving action of a noncompact, simply connected, almost simple Lie group having Kazhdan’s property (T). The technique of proof also shows that M does not admit a hyperbolic structure.

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Authors and Affiliations

  1. Department of Mathematics, University of Chicago, 60637, Chicago, IL, USA

    Scot Adams

  2. Department of Mathematics, Indiana University, 47405, Bloomington, IN, USA

    Russell Lyons

Authors
  1. Scot Adams
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  2. Russell Lyons
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Additional information

The research of both authors was supported by NSF Postdoctoral Fellowships. The second author was also supported by an Alfred P. Sloan Foundation Research Fellowship.

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Adams, S., Lyons, R. Amenability, Kazhdan’s property and percolation for trees, groups and equivalence relations. Israel J. Math. 75, 341–370 (1991). https://doi.org/10.1007/BF02776032

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  • DOI: https://doi.org/10.1007/BF02776032

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