Triangulations of 3-manifolds, hyperbolic relative handlebodies, and Dehn filling

  • Francesco Costantino

    Université Paul Sabatier, Toulouse, France
  • Roberto Frigerio

    Università di Pisa, Italy
  • Bruno Martelli

    Università di Pisa, Italy
  • Carlo Petronio

    Università di Pisa, Italy


We establish a bijective correspondence between the set of 3-dimensional triangulations with tetrahedra and a certain class of relative handlebodies (i.e. handlebodies with boundary loops, as defined by Johannson) of genus .

We show that the manifolds in are hyperbolic (with geodesic boundary, and cusps corresponding to the loops), have least possible volume, and simplest boundary loops.

Mirroring the elements of in their geodesic boundary we obtain a set of cusped hyperbolic manifolds, previously considered by . Thurston and the first named author. We show that also corresponds bijectively to , and we study some Dehn fillings of the manifolds in . As consequences of our constructions, we also show that:

  • A triangulation of a 3-manifold is uniquely determined up to isotopy by its 1-skeleton;

  • If a 3-manifold has an ideal triangulation with edges of valence at least 6, then is hyperbolic and the edges are homotopically non-trivial, whence homotopic to geodesics;

  • Every finite group is the isometry group of a closed hyperbolic 3-manifold with volume less than .

Cite this article

Francesco Costantino, Roberto Frigerio, Bruno Martelli, Carlo Petronio, Triangulations of 3-manifolds, hyperbolic relative handlebodies, and Dehn filling. Comment. Math. Helv. 82 (2007), no. 4, pp. 903–933

DOI 10.4171/CMH/114