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Homology and volume of hyperbolic 3-orbifolds, and enumeration of arithmetic groups
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 24 (2015) no. 5, pp. 1147-1156.

Selon un théorème de Borel, les volumes d’orbifolds hyperboliques arithmétiques de dimension 3 constituent un ensemble discret. Ce théorème soulève le problème d’énumérer les orbifolds hyperboliques arithmétiques de dimension 3 dont le volume est majoré par une constante donnée. Une étape cruciale dans ce programme est de majorer le rang d’un certain 2-groupe abélien élémentaire associé à un tel orbifold O. Ce rang est majoré par la dimension de H 1 (O; 2 ). Étant donné une variété hyperbolique M dont le volume est majoré par une constante convenable, des résultats que j’ai établis en collaboration avec Marc Culler et d’autres auteurs donnent des bornes supérieures utiles pour la dimension de H 1 (M; 2 ). Dans cet article je décrirai mes progrès sur le problème d’étendre les résultats de ce genre au cadre des orbifolds.

Borel’s theorem that volumes of arithmetic hyperbolic 3-orbifolds form a discrete set raises the problem of enumerating those arithmetic hyperbolic 3-orbifolds whose volume is subject to a given upper bound. A key step is bounding the rank of a certain elementary abelian 2-group associated with such an orbifold O. This rank is bounded above by the dimension of H 1 (O; 2 ). Joint work of mine with Marc Culler and others gives good bounxds for the dimension of H 1 (M; 2 ), where M is a hyperbolic 3-manifold whose volume has a suitable upper bound. I will report on progress on the problem of extending results of this kind to orbifolds.

Publié le :
DOI : https://doi.org/10.5802/afst.1478
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Peter B. Shalen. Homology and volume of hyperbolic $3$-orbifolds, and enumeration of arithmetic groups. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 24 (2015) no. 5, pp. 1147-1156. doi : 10.5802/afst.1478. https://afst.centre-mersenne.org/articles/10.5802/afst.1478/

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