Thesis advisor: Martin Bridgeman / We begin this dissertation by studying the relationship between the Poincaré metric of a simply connected domain Ω ⊂ ℂ and the geometry of Dome(Ω), the boundary of the convex hull of its complement. Sullivan showed that there is a universal constant K[subscript]eq[subscript] such that one may find a conformally natural K[subscript]eq[subscript]-quasiconformal map from Ω to Dome(Ω) which extends to the identity on ∂Ω. Explicit upper and lower bounds on K[subscript]eq[subscript] have been obtained by Epstein, Marden, Markovic and Bishop. We improve upon these upper bounds by showing that one may choose K[subscript]eq[subscript] ≤ 7.1695. As part of this work, we provide stronger criteria for embeddedness of pleated planes. In addition, for Kleinian groups Γ where N = ℍ³/Γ has incompressible boundary, we give improved bounds for the average bending on the convex core of N and the Lipschitz constant for the homotopy inverse of the nearest point retraction. In the second part of this dissertation, we prove an extension of Basmajian's identity to n-Hitchin representations of compact bordered surfaces. For 3-Hitchin representations, we provide a geometric interpretation of this identity analogous to Basmajian's original result. As part of our proof, we demonstrate that for a closed surface, the Lebesgue measure on the Frenet curve of an n-Hitchin representation is zero on the limit set of any incompressible subsurface. This generalizes a classical result in hyperbolic geometry. In our final chapter, we prove the Bridgeman-Kahn identity for all finite volume hyperbolic n-manifolds with totally geodesic boundary. As part of this work, we correct a commonly referenced expression of the volume form on the unit tangent bundle of ℍⁿ in terms of the geodesic end point parametrization. / Thesis (PhD) — Boston College, 2016. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

Thesis advisor: Martin J. Bridgeman / Thesis advisor: Ian Biringer / The first part of this dissertation is on the quasiconformal homogeneity of surfaces. In the vein of Bonfert-Taylor, Bridgeman, Canary, and Taylor we introduce the notion of quasiconformal homogeneity for closed oriented hyperbolic surfaces restricted to subgroups of the mapping class group. We find uniform lower bounds for the associated quasiconformal homogeneity constants across all closed hyperbolic surfaces in several cases, including the Torelli group, congruence subgroups, and pure cyclic subgroups. Further, we introduce a counting argument providing a possible path to exploring a uniform lower bound for the nonrestricted quasiconformal homogeneity constant across all closed hyperbolic surfaces. We then move on to identities on hyperbolic manifolds. We study the statistics of the unit geodesic flow normal to the boundary of a hyperbolic manifold with non-empty totally geodesic boundary. Viewing the time it takes this flow to hit the boundary as a random variable, we derive a formula for its moments in terms of the orthospectrum. The first moment gives the average time for the normal flow acting on the boundary to again reach the boundary, which we connect to Bridgeman's identity (in the surface case), and the zeroth moment recovers Basmajian's identity. Furthermore, we are able to give explicit formulae for the first moment in the surface case as well as for manifolds of odd dimension. In dimension two, the summation terms are dilogarithms. In dimension three, we are able to find the moment generating function for this length function. / Thesis (PhD) — Boston College, 2015. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.