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- Carlo Petronio, Joan Porti
- 2008

We give a complete proof of Thurston's celebrated hyperbolic Dehn filling theorem , following the ideal triangulation approach of Thurston and Neumann-Zagier. We avoid to assume that a genuine ideal triangulation always exists, using only a partially flat one, obtained by subdividing an Epstein-Penner decomposition. This forces us to deal with negatively… (More)

If Σ → Σ is a branched covering between closed surfaces, there are several easy relations one can establish between χ(Σ), χ(Σ), orientability of Σ and Σ, the total degree, and the local degrees at the branching points, including the classical Riemann-Hurwitz formula. These necessary relations have been shown to be also sufficient for the existence of the… (More)

- Bruno Martelli, Carlo Petronio
- Experimental Mathematics
- 2001

- Roberto Frigerio, Bruno Martelli, Carlo Petronio
- Experimental Mathematics
- 2004

We classify the orientable finite-volume hyperbolic 3-manifolds having non-empty compact totally geodesic boundary and admitting an ideal triangula-tion with at most four tetrahedra. We also compute the volume of all such manifolds, we describe their canonical Kojima decomposition, and we discuss manifolds having cusps. The manifolds built from one or two… (More)

We compute for all orientable irreducible geometric 3-manifolds certain complexity functions that approximate from above Matveev's natural complexity, known to be equal to the minimal number of tetrahedra in a triangulation. We can show that the upper bounds on Matveev's complexity implied by our computations are sharp for thousands of manifolds, and we… (More)

For the existence of a branched covering Σ → Σ between closed surfaces there are easy necessary conditions in terms of χ(Σ), χ(Σ), orientability, the total degree, and the local degrees at the branching points. A classical problem dating back to Hurwitz asks whether these conditions are also sufficient. Thanks to the work of many authors, the problem… (More)

We classify all the non-hyperbolic Dehn fillings of the complement of the chain-link with 3 components, deducing the classification of all non-hyperbolic Dehn fillings of infinitely many 1-cusped and 2-cusped hyperbolic manifolds, including many of the smallest ones known. We study in this paper the Dehn fillings of the complement N of the chain-link with 3… (More)

- Carlo Petronio
- 2008

The famous Haken-Kneser-Milnor theorem states that every 3-manifold can be expressed in a unique way as a connected sum of prime 3-manifolds. The analogous statement for 3-orbifolds has been part of the folklore for several years, and it was commonly believed that slight variations on the argument used for manifolds would be sufficient to establish it. We… (More)

We define for each g 2 and k 0 a set M g,k of orientable hyperbolic 3-manifolds with k toric cusps and a connected totally geodesic boundary of genus g. Manifolds in M g,k have Matveev complexity g + k and Heegaard genus g + 1, and their homology, volume, and Turaev-Viro invariants depend only on g and k. In addition, they do not contain closed essential… (More)

Using the theory of hyperbolic manifolds with totally ge-odesic boundary, we provide for every n 2 a class M n of such manifolds all having Matveev complexity equal to n and Heegaard genus equal to n + 1. All the elements of M n have a single boundary component of genus n, and #M n grows at least exponentially with n. This paper is devoted to the… (More)