Related papers: Commensurators of cusped hyperbolic manifolds

In this paper we enumerate and classify the ``simplest'' pairs (M,G) where M is a closed orientable 3-manifold and G is a trivalent graph embedded in M. To enumerate the pairs we use a variation of Matveev's definition of complexity for…

Thurston conjectured that a closed triangulated 3-manifold in which every edge has degree 5 or 6, and no two edges of degree 5 lie in a common 2-cell, has word-hyperbolic fundamental group. We establish Thurston's conjecture by proving that…

Every cusped, finite-volume hyperbolic three-manifold has a canonical decomposition into ideal polyhedra. We study the canonical decomposition of the hyperbolic manifold obtained by filling some (but not all) of the cusps with solid tori:…

Given a triangulation of a closed, oriented, irreducible, atoroidal 3-manifold every oriented, incompressible surface may be isotoped into normal position relative to the triangulation. Such a normal oriented surface is then encoded by…

This paper employs knot invariants and results from hyperbolic geometry to develop a practical procedure for checking the cosmetic surgery conjecture on any given one-cusped manifold. This procedure has been used to establish the following…

It is conjectured that every cusped hyperbolic 3-manifold admits a geometric triangulation, i.e. it is decomposed into positive volume ideal hyperbolic tetrahedra. Here, we show that sufficiently highly twisted knots admit a geometric…

We outline a rigorous algorithm, first suggested by Casson, for determining whether a closed orientable 3-manifold M is hyperbolic, and to compute the hyperbolic structure, if one exists. The algorithm requires that a procedure has been…

Following on from work of Dunfield, we determine the fibred status of all the unknown hyperbolic 3-manifolds in the cusped census. We then find all the fibred hyperbolic 3-manifolds in the closed census and use this to find over 100…

In this article we study the spectrum of totally geodesic surfaces of a finite volume hyperbolic 3-manifold. We show that for arithmetic hyperbolic 3-manifolds that contain a totally geodesic surface, this spectrum determines the…

From its creation in 1989 through subsequent extensions, the widely-used "SnapPea census" now aims to represent all cusped finite-volume hyperbolic 3-manifolds that can be obtained from <= 8 ideal tetrahedra. Its construction, however, has…

This paper introduces a rigorous computer-assisted procedure for analyzing hyperbolic 3-manifolds. This technique is used to complete the proof of several long-standing rigidity conjectures in 3-manifold theory as well as to provide a new…

We classify the topological types for the unions of the totally geodesic 3-punctured spheres in orientable hyperbolic 3-manifolds. General types of the unions appear in various hyperbolic 3-manifolds. Each of the special types of the unions…

Given a cusped hyperbolic 3-manifold with finite volume, we define two types of complex parameters which capture geometric information about the preimages of geodesic arcs traveling between cusp cross-sections. We prove that these…

Three dimensional hyperbolic manifolds have accumulation points in the spectrum of their volumes, leading to a divergence in the sum over topologies. The limit points are cusped hyperbolic manifolds, and we propose to renormalize the sum by…

By extending the concept of Euler-angle rotations to more than three dimensions, we develop the systematics under rotations in higher-dimensional space for a novel set of hyperspherical harmonics. Applying this formalism, we determine all…

L. Paoluzzi constructed a family of compact orientable three-dimensional hyperbolic manifolds with totally geodesic boundary, which were, by construction, closely related to the three-dimensional torus. This paper gives their complete…

A multi-cube method is developed for solving systems of elliptic and hyperbolic partial differential equations numerically on manifolds with arbitrary spatial topologies. It is shown that any three-dimensional manifold can be represented as…

Assuming that every hyperbolic group is residually finite, we prove the congruence subgroup property for mapping class groups of hyperbolic surfaces of finite type. Under the same assumption, it follows that profinitely equivalent…

We show that a hyperbolic 2-bridge knot complement is the unique knot complement in its commensurability class. We also discuss constructions of commensurable hyperbolic knot complements and put forth a conjecture on the number of…

This thesis investigates cusp cross-sections of arithmetic real, complex, and quaternionic hyperbolic $n$--orbifolds. We give a smooth classification of these submanifolds and analyze their induced geometry. One of the primary tools is a…