15 February 2015 Cheeger constants and L2-Betti numbers
Lewis Bowen
Duke Math. J. 164(3): 569-615 (15 February 2015). DOI: 10.1215/00127094-2871415


We prove the existence of positive lower bounds on the Cheeger constants of manifolds of the form X/Γ where X is a contractible Riemannian manifold and Γ<Isom(X) is a discrete subgroup, typically with infinite covolume. The existence depends on the L2-Betti numbers of Γ, its subgroups, and a uniform lattice of Isom(X). As an application, we show the existence of a uniform positive lower bound on the Cheeger constant of any manifold of the form H4/Γ where H4 is real hyperbolic 4-space and Γ<Isom(H4) is discrete and isomorphic to a subgroup of the fundamental group of a complete finite-volume hyperbolic 3-manifold. Via Patterson–Sullivan theory, this implies the existence of a uniform positive upper bound on the Hausdorff dimension of the conical limit set of such a Γ when Γ is geometrically finite. Another application shows the existence of a uniform positive lower bound on the zeroth eigenvalue of the Laplacian of Hn/Γ over all discrete free groups Γ<Isom(Hn) whenever n4 is even. (The bound depends on n.) This extends results of Phillips–Sarnak and Doyle, who obtained such bounds for n3 when Γ is a finitely generated Schottky group.


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Lewis Bowen. "Cheeger constants and L2-Betti numbers." Duke Math. J. 164 (3) 569 - 615, 15 February 2015. https://doi.org/10.1215/00127094-2871415


Published: 15 February 2015
First available in Project Euclid: 17 February 2015

zbMATH: 1312.57041
MathSciNet: MR3314481
Digital Object Identifier: 10.1215/00127094-2871415

Primary: 22F30
Secondary: 57S30

Keywords: $L2$ Betti numbers , Cheeger constant , hyperbolic manifold

Rights: Copyright © 2015 Duke University Press

Vol.164 • No. 3 • 15 February 2015
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