Sur le volume des simplexes hyperboliques idéaux

Abstract : The volume of hyperbolic simplicies plays an important role in the investigation of the volume of hyperbolic manifolds and in other fields of mathematics such that arithmetic. But it is far more difficult to compute than eucildean volume. In this thesis, we generalize a formula used by A. Connes to compute the area of triangles in euclidean or hyperbolic spaces: we show a formula giving the volume of finite and ideals hyperbolic simplicies. However, this formula does not allow straightforward computations. That is why we only use it in order to get analytic properties of the volume function of ideal simplicies. More precisely, seeing an ideal simplex in the Poincaré ball model or the Klein model of the hyperbolic space, we can consider its volume as a function of its vertices on a sphere. After giving a basis of spherical harmonics on the sphere, we set out a method for expanding the volume function of ideal simplicies in spherical harmonics. As an example, we make the computations for dimensions 2 and 3. But even in these cases, it does not lead to simple formualae for the coefficients of the expansion. That is why we use the software Maple in order to compute the "little" ones.
Document type :
Theses
Mathematics [math]. Université Joseph-Fourier - Grenoble I, 2001. French


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Submitted on : Tuesday, January 8, 2002 - 7:53:05 AM
Last modification on : Tuesday, January 8, 2002 - 7:53:05 AM

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Richard Pereyrol. Sur le volume des simplexes hyperboliques idéaux. Mathematics [math]. Université Joseph-Fourier - Grenoble I, 2001. French. <tel-00000956>

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