Geodesic
Uniform non-amenability, cost, and the first $\ell^2$-Betti number
Russell Lyons1 ; Mikaël Pichot2 ; Stéphane Vassout3
1Indiana University, Bloomington, United States
2IHES, Bures-Sur-Yvette, France
3Institut de Mathématiques de Jussieu - Paris Rive Gauche, France
Groups, geometry, and dynamics, Tome 2 (2008) no. 4, pp. 595-617

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DOI

It is shown that 2β1(Γ) ≤ h(Γ) for any countable group Γ, where β1(Γ) is the first l2-Betti number and h(Γ) the uniform isoperimetric constant. In particular, a countable group with non-vanishing first l2-Betti number is uniformly non-amenable. We then define isoperimetric constants in the framework of measured equivalence relations. For an ergodic measured equivalence relation R of type II_1, the uniform isoperimetric constant h(R) of R is invariant under orbit equivalence and satisfies
DOI : 10.4171/ggd/49
Classification : 20-XX, 00-XX
Mots-clés : l<sup>2</sup>-Betti numbers, uniform non-amenability, measured equivalence relations

Russell Lyons  1   ; Mikaël Pichot  2   ; Stéphane Vassout  3

1 Indiana University, Bloomington, United States
2 IHES, Bures-Sur-Yvette, France
3 Institut de Mathématiques de Jussieu - Paris Rive Gauche, France 
Russell Lyons; Mikaël Pichot; Stéphane Vassout. Uniform non-amenability, cost, and the first $\ell^2$-Betti number. Groups, geometry, and dynamics, Tome 2 (2008) no. 4, pp. 595-617. doi: 10.4171/ggd/49
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     title = {Uniform non-amenability, cost, and the first $\ell^2${-Betti} number},
     journal = {Groups, geometry, and dynamics},
     pages = {595--617},
     year = {2008},
     volume = {2},
     number = {4},
     doi = {10.4171/ggd/49},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/49/}
}
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