Euclidean Structures on Simplicial Surfaces and Hyperbolic Volume

  title={Euclidean Structures on Simplicial Surfaces and Hyperbolic Volume},
  author={Igor Rivin},
  journal={Annals of Mathematics},
  • Igor Rivin
  • Published 1 May 1994
  • Mathematics
  • Annals of Mathematics
In this paper we characterize the edge invariant and Delaunay invariant of a spherical angle structure on a triangulated surface. We also characterize the edge invariant of a hyperbolic angle
Geometric triangulations of a family of hyperbolic 3-braids
We construct topological triangulations for complements of (−2, 3, n)-pretzel knots and links with n ≥ 7. Following a procedure outlined by Futer and Guéritaud, we use a theorem of Casson and Rivin
Geometric triangulations and highly twisted links
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
Geometrically and diagrammatically maximal knots
It is shown that many families of alternating knots and links simultaneously maximize both ratios of the ratio of volume to crossing number and the knot determinant per crossing.
Discrete conformal maps and ideal hyperbolic polyhedra
We establish a connection between two previously unrelated topics: a particular discrete version of conformal geometry for triangulated surfaces, and the geometry of ideal polyhedra in hyperbolic
Delaunay triangulations with disconnected realization spaces
In general, it is proved that the realization space of a Delaunay triangulation in Rd can have Ω(2d) connected components and also shows that the realizability problem for Delauny triangulations is polynomially equivalent to the existential theory of the reals.
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
Rigidity of Polyhedral Surfaces, III
This paper investigates several global rigidity issues for polyhedral surfaces including inversive distance circle packings. Inversive distance circle packings are polyhedral surfaces introduced by
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
In this article, we describe a method for computing generators of the Veech group of a at surface (which we dene as a Riemann surface with a non-zero holomorphic quadratic dierential). The method


Geometry of Spaces of Constant Curvature
This paper develops elementary geometry in a way very similar to that used to create the geometry the authors learned at school, but since its basic notions can be interpreted in different ways, this geometry can be applied to objects other than the conventional physical space.
A characterization of convex hyperbolic polyhedra and of convex polyhedra inscribed in the sphere
We describe a characterization of convex polyhedra in H 3 in terms of their dihedral angles, developed by Rivin. We also describe some geometric and combinatorial consequences of that theory. One of