Spectral theory, Hausdorff dimension and the topology of hyperbolic 3-manifolds

Abstract

Let M be a compact 3-manifold whose interior admits a complete hyperbolic structure. We let Λ(M ) be the supremum of λ 0 (N ) where N varies over all hyperbolic 3-manifolds homeomorphic to the interior of M . Similarly, we let D(M ) be the infimum of the Hausdorff dimensions of limit sets of Kleinian groups whose quotients are homeomorphic to the interior of M . We observe that Λ(M ) = D(M )(2 − D(M )) if M is not handlebody or a thickened torus. We characterize exactly when Λ(M ) = 1 and D(M ) = 1 in terms of the characteristic submanifold of the incompressible core of M .

Boundary conformal field theories, limit sets of Kleinian groups and holography

“…In the case if hyperbolic manifold M 3 is topologically tame (that is it is homeomorphic to the interior of a compact 3 manifold), then Theorem 2.1. of Canary et all [79] states that Theorem 6.3. (Canary, Minsky and Taylor) If M 3 is topologically tame hyperbolic 3-manifold, then the lowest eigenvalue λ 0 of the hyperbolic Laplacian (-∆ h ) is given by λ 0 = α(2−α) unless α < 1, in which case λ 0 (M 3 ) = 1.…”

An introduction to pressure metrics for higher Teichmüller spaces

“…It is conjectured that the entropy has a unique minimum at this representation (see Canary-Minsky-Taylor [23]). Storm [80] proved that this is the unique representation where the volume of the convex core achieves its minimum.…”

On the spectral theory and dynamics of asymptotically hyperbolic manifolds

“…Specifically, there are examples of convex co-compact hyperbolic manifolds whose geodesic flow has topological entropy h > n 2 ; see for example Canary, Minsky, and Taylor [19] who construct examples of hyperbolic 3-manifolds. By the prime orbit theorem for these manifolds [76], Lemma 4.3, and a straightforward estimate, the renormalized wave trace lies only in D ((0, ∞)) rather than S ((0, ∞)).…”