Martin Bridgeman

From the Boundary of the Convex Core to the Conformal Boundary

If N is a hyperbolic 3-manifold with finitely generated fundamental group, then the nearest point retraction is a proper homotopy equivalence from the conformal boundary of N to the boundary of the convex core of N. We show that the nearest point retraction is Lipschitz and has a Lipschitz homotopy inverse and that one may bound the Lipschitz constants in terms of the length of the shortest compressible curve on the conformal boundary.

Hyperbolic Volume of Manifolds with Geodesic Boundary and Orthospectra

In this paper we describe a function Fn : R+ → R+ such that for any hyperbolic n-manifold M with totally geodesic boundary

Distribution of intersection lengths of a random geodesic with a geodesic lamination

{\partial M \neq \emptyset}

Average curvature of convex curves in

, the volume of M is equal to the sum of the values of Fn on the orthospectrum of M. We derive an integral formula for Fn in terms of elementary functions. We use this to give a lower bound for the volume of a hyperbolic n-manifold with totally geodesic boundary in terms of the area of the boundary.

Length distortion and the Hausdorff dimension of limit sets

We show that any closed hyperbolic surface admitting a conformal automorphism with ?many? fixed points is uniformly quasiconformally homogeneous, with constant uniformly bounded away from 1. In particular, there is a uniform lower bound on the quasiconformal homogeneity constant for all hyperelliptic surfaces. In addition, we introduce more restrictive notions of quasiconformal homogeneity and bound the associated quasiconformal homogeneity constants uniformly away from 1 for all hyperbolic surfaces.

The Structure and Enumeration of Link Projections

We investigate the distribution of lengths obtained by intersecting a random geodesic with a geodesic lamination. We give an explicit formula for the distribution for the case of a maximal lamination and show that the distribution is independent of the surface and lamination. We also show how the moments of the distribution are related to the Riemann zeta function.

Simple root flows for Hitchin representations

In this paper we consider finite volume hyperbolic manifolds X with non-empty totally geodesic boundary. We consider the distribution of the times for the geodesic flow to hit the boundary and derive a formula for the moments of the associated random variable in terms of the orthospectrum. We show that the the first two moments correspond to two cases of known identities for the orthospectrum. We further obtain an explicit formula in terms of the trilogarithm functions for the average time for the geodesic flow to hit the boundary in the surface case, using the third moment.

An improved bound for Sullivan's convex hull theorem

A well-known result states that, if a curve a in H2 has geodesic curvature less than or equal to one at every point, then a is embedded. The converse is obviously not true, but the embeddedness of a curve does give information about the curvature. We prove that, if a is a convex embedded curve in H2, then the average curvature (curvature per unit length) of a, denoted K(a), satisfies K(a) < 1. This bound on the average curvature is tight as K(a) = 1 for a a horocycle.