1,669 research outputs found
Cheeger constants and -Betti numbers
We prove the existence of positive lower bounds on the Cheeger constants of
manifolds of the form where is a contractible Riemannian
manifold and \Gamma<\Isom(X) is a discrete subgroup, typically with infinite
co-volume. The existence depends on the -Betti numbers of , its
subgroups and of 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 \H^4/\Gamma where \H^4 is real hyperbolic 4-space and
\Gamma<\Isom(\H^4) 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
when is geometrically finite. Another application shows the
existence of a uniform positive lower bound on the zero-th eigenvalue of the
Laplacian of \H^n/\Gamma over all discrete free groups \Gamma<\Isom(\H^n)
whenever is even (the bound depends on ). This extends results of
Phillips-Sarnak and Doyle who obtained such bounds for when
is a finitely generated Schottky group.Comment: Comments welcome. This new version corrects a few minor error
Spectral and Hodge theory of `Witt' incomplete cusp edge spaces
Incomplete cusp edges model the behavior of the Weil-Petersson metric on the compactified Riemann moduli space near the interior of a divisor. Assuming such a space is Witt, we construct a fundamental solution to the heat equation, and using a precise description of its asymptotic behavior at the singular set, we prove that the Hodge-Laplacian on differential forms is essentially self-adjoint, with discrete spectrum satisfying Weyl asymptotics. We go on to prove bounds on the growth of -harmonic forms at the singular set and to prove a Hodge theorem, namely that the space of -harmonic forms is naturally isomorphic to the middle-perversity intersection cohomology. Moreover, we develop an asymptotic expansion for the heat trace near
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