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Pré-Publication, Document De Travail Année : 2020

Quantitative measure equivalence between finitely generated groups

Résumé

We initiate a quantitative study of measure equivalence (and orbit equivalence) between finitely generated groups that extends the classical setting of L p measure equivalence. In particular we investigate quantitative versions of Orstein Weiss' theorem. Our first main result is a very general monotonicity property satisfied by the isoperimetric profile, which provides a "lower bound" on how integrable a measure coupling between Z and an amenable group Γ can be. As another application we get that the isoperimetric profile is invariant under L 1 measure equivalence and monotonous under coarse embedding among amenable groups. On the other hand, we introduce a notion of Følner tiling, which we use to construct explicit orbit equivalences between amenable groups satisfying certain integrability conditions. We show in a number of instances that the bound given by the isoperimetric profile are sharp up to a logarithmic error. In the context of non-amenable groups, we study the stability of hyperbolicity: Shalom proved that Gromov hyperbolicity is not invariant under L p measure equivalence. Actually his proof yields a stronger statement, which we prove to be optimal in some sense. As a by-product, we show that if Γ and Λ admit an orbit equivalence coupling satisfying a certain integrability condition, and if Γ is hyperbolic, then so is Λ.
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Dates et versions

hal-02465155 , version 1 (03-02-2020)

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  • HAL Id : hal-02465155 , version 1

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Thiebout Delabie, Juhani Koivisto, François Le Maître, Romain Tessera. Quantitative measure equivalence between finitely generated groups. 2020. ⟨hal-02465155⟩
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