From angled triangulations to hyperbolic structures

  title={From angled triangulations to hyperbolic structures},
  author={David Futer and Franccois Gu'eritaud},
  journal={arXiv: Geometric Topology},
This survey paper contains an elementary exposition of Casson and Rivin's technique for finding the hyperbolic metric on a 3-manifold M with toroidal boundary. We also survey a number of applications of this technique. The method involves subdividing M into ideal tetrahedra and solving a system of gluing equations to find hyperbolic shapes for the tetrahedra. The gluing equations decompose into a linear and non-linear part. The solutions to the linear equations form a convex polytope A. The… 

Figures from this paper

Computing complete hyperbolic structures on cusped 3-manifolds
This paper proposes a new method to find a triangulation that admits a solution to the gluing equations, using convex optimization and combinatorial modifications, based on Casson and Rivin's reformulation of the equations.
A combinatorial curvature flow for ideal triangulations
We investigate a combinatorial analogue of the Ricci curvature flow for 3-dimensional hyperbolic cone structures, obtained by gluing together hyperbolic ideal tetrahedra. Our aim is to find a
Hyperbolic Knot Theory
This book is an introduction to hyperbolic geometry in dimension three, and its applications to knot theory and to geometric problems arising in knot theory. It has three parts. The first part covers
An Introduction to Geometric Topology
This book provides a self-contained introduction to the topology and geometry of surfaces and three-manifolds. The main goal is to describe Thurston's geometrisation of three-manifolds, proved by
Geometric triangulations and the Teichm\"uller TQFT volume conjecture for twist knots.
We construct a new infinite family of ideal triangulations and H-triangulations for the complements of twist knots, using a method originating from Thurston and Kashaev-Luo-Vartanov. These
Pseudo-developing maps for ideal triangulations II: Positively oriented ideal triangulations of cone-manifolds
We generalise work of Young-Eun Choi to the setting of ideal triangulations with vertex links of arbitrary genus, showing that the set of all (possibly incomplete) hyperbolic cone-manifold structures
Combinatorial Ricci flows and the hyperbolization of a class of compact 3-manifolds
We prove that for a compact 3-manifold M with boundary admitting an ideal triangulation T with valence at least 10 at all edges, there exists a unique complete hyperbolic metric with totally geodesic
Infinitely many virtual geometric triangulations
We prove that every cusped hyperbolic 3–manifold has a finite cover admitting infinitely many geometric ideal triangulations. Furthermore, every long Dehn filling of one cusp in this cover admits
Combinatorial Ricci Flows with Applications to the Hyperbolization of Cusped 3-Manifolds
In this paper, we adopt combinatorial Ricci flow to study the existence of hyperbolic structure on cusped 3-manifolds. The long-time existence and the uniqueness for the extended combinatorial
A note on complete hyperbolic structures on ideal triangulated 3-manifolds
It is a theorem of Casson and Rivin that the complete hyperbolic metric on a cusp end ideal triangulated 3-manifold maximizes volume in the space of all positive angle structures. We show that the


Volume Optimization, Normal Surfaces and Thurston's Equation on Triangulated 3-Manifolds
We propose a finite dimensional variational principle on triangulated 3-manifolds so that its critical points are related to solutions to Thurston's gluing equation and Haken's normal surface
Triangulated 3-Manifolds: from Haken's normal surfaces to Thurston's algebraic equation
We give a brief summary of some of our work and our joint work with Stephan Tillmann on solving Thurston's equation and Haken equation on triangulated 3-manifolds in this paper. Several conjectures
Cusp Areas of Farey Manifolds and Applications to Knot Theory
This paper gives the first explicit, two-sided estimates on the cusp area of once-punctured-torus bundles, 4-punctured sphere bundles, and two-bridge link complements. The input for these estimates
Combinatorics of Triangulations and the Chern-Simons Invariant for Hyperbolic 3-Manifolds
In this paper we prove some results on combinatorics of triangulations of 3-dimensional pseudo-manifolds, improving on results of [NZ], and apply them to obtain a simplicial formula for the
Canonical triangulations of Dehn fillings
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
Volumes of hyperbolic three-manifolds
A characterization of ideal polyhedra in hyperbolic $3$-space
The goals of this paper are to provide a characterization of dihedral angles of convex ideal (those with all vertices on the sphere at infinity) polyhedra in H3, and also of those convex polyhedra
Triangulated cores of punctured-torus groups
We show that the interior of the convex core of a quasifuchsian punctured-torus group admits an ideal decomposition (usually an infinite triangulation) which is canonical in two different senses: in
Angled decompositions of arborescent link complements
This paper describes a way to subdivide a 3‐manifold into angled blocks, namely polyhedral pieces that need not be simply connected. When the individual blocks carry dihedral angles that fit together