Arnold's seminar, Moscow University
March 11, 2003

A.D.Mednykh (Novosibirsk)
"Volumes of hyperbolic 3-manifolds  and orbifolds"

By the Thurston-Jorgensen theorem, the volumes of hyperbolic
3-dimensional manifolds and orbifolds form on the real line
a well-ordrered subset ot the type
$\omega^\omega$.   The calculation of the volumes of 3-manifolds is a very
interesting and difficult problem.   First of all we show that known
hyperbolic manifolds of small volume are cyclic coverings of the
3-sphere branched over a link or a knot. Then we use the geometrical
properties of the orbifold obtained in this way for volume
calculation.  Then we emphasize that the volumes of orbifolds are
closely related to volumes of polyhedra
in the hyperbolic space and volumes of convex hulls
for quasifuchsian groups and fractal sets.
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