Critical Exponent of Negatively Curved Three Manifolds

Abstract

Abstract. We prove that for a negatively pinched (−b 2 ≤ K ≤ −1) topologically tame 3-manifoldM/Γ , all geometrically infinite ends are simply degenerate. And if the limit set of Γ is the entire boundary sphere at infinity, then the action of Γ on the boundary sphere is ergodic with respect to harmonic measure, and the Poincaré series diverges when the critical exponent is 2.2000 Mathematics Subject Classification. 57M50, 57M60.

Bounded cohomology and negatively curved manifolds

“…If M has boundary-irreducible compact core, then every geometrically infinite end of M is simply degenerate. This result is also available without the assumption that M has boundaryirreducible compact core, which was proved by Hou [20]. Therefore, if M is not geometrically finite, M has a simply degenerate end.…”

“…By the works of Bonahon [3] and Hou [20], the conditions (a) and (b) are equivalent. Sikorav [28] shows that the conditions (b) and (c) are equivalent if M has bounded geometry in the sense that it is complete, its sectional curvature is bounded in absolute value, and its injectivity radius is bounded below.…”

Pants immersed in hyperbolic 3-manifolds

“…He also showed (generalizing an argument of Thurston [55]) that geometric tameness implies the Ahlfors measure conjecture [3]. Canary's arguments were generalized for PNC manifolds by Yong Hou [34]. In fact, these arguments give a geometric proof of the Ahlfors finiteness theorem [3].…”

“…Then the sequence of lifts of geodesics β * i must exit the end F of N ′ corresponding to the compressible component of ∂C. Thus, the compressible end F of N ′ is geometrically infinite, and is therefore simply degenerate by [18,34]. By the PNC covering theorem 14.2, the end F of N must cover finite-to-one an end E of M α * (2, 0).…”

“…Perturb the metric on M near α * to a PNC metric ν ′ such that α * is homotopic to an embedded geodesic α ′ , by the method of Lemma 5.5, [18]. Then we may deform the metric near α ′ to a metric ν ′′ so that the metric near α ′′ (the geodesic representative of α in ν ′′ ) is locally isometric to H 3 b , b < 0, by Lemma 3.2 and 4.2, [34]. To simplify notation, we will denote this new metric by ν, and we will assume that the geodesic representative of α is a geodesic link α * with neighborhood locally isometric to…”