Carlo Petronio

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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)
This paper is devoted to the investigation of the class Mn of orientable compact 3-manifolds having an ideal triangulation with n 2 tetrahedra and a single edge. We show in particular for each M in Mn that the Heegaard genus of M is equal to n+1, and that the complexity ofM in the sense of Matveev is equal to n. Moreover we prove that the classical(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. MSC (2000): 57M27 (primary), 57M20, 57M50 (secondary). We study in this paper the(More)
The famous Haken-Kneser-Milnor theorem states that every 3-manifold can be expressed in a unique way as a connected sum of prime 3manifolds. 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 classify the orientable finite-volume hyperbolic 3-manifolds having nonempty compact totally geodesic boundary and admitting an ideal triangulation 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)
Given a branched covering between closed connected surfaces, one can easily establish some relations between the Euler characteristic and orientability of the involved surfaces, the degree of the covering, the number of branching points, and the local degrees at these points. These relations can therefore be regarded as necessary conditions for the(More)
We extend Matveev’s theory of complexity for 3-manifolds, based on simple spines, to (closed, orientable, locally orientable) 3-orbifolds. We prove naturality and finiteness for irreducible 3-orbifolds, and, with certain restrictions and subtleties, additivity under orbifold connected sum. We also develop the theory of handle decompositions for 3orbifolds(More)
Let Σ̃ and Σ be closed, connected, and orientable surfaces, and let f : Σ̃ → Σ be a branched cover. For each branching point x ∈ Σ the set of local degrees of f at f−1(x) is a partition of the total degree d. The total length of the various partitions is determined by χ(Σ̃), χ(Σ), d and the number of branching points via the Riemann-Hurwitz formula. A very(More)