Geometry, Topology and Dynamics Seminar
Carlo Petronio
University of Pisa
ON THE HURWITZ EXISTENCE PROBLEM FOR
Abstract: Branched coverings between closed surfaces, or equivalently orbifold
coverings between 2-orbifolds, naturally arise when one considers
coverings between closed Seifert 3-manifolds. Using conjectural formulae
of Martelli and the speaker, Seifert coverings can be used to investigate
the behaviour of Matveev's complexity for 3-manifolds under general finite
coverings. With this in mind, Pervova and the speaker have considered the
very old problem, basically dating back to Hurwitz, of the existence of a
branched covering between given closed surfaces with given branch data.
Local compatibility and the Riemann-Hurwitz formula give necessary
conditions for existence. Thanks to the work of several authors over
the decades, these conditions are now known to be also sufficient, except for
the case where the base surface is the sphere. In this case exceptions
are known to exist, and the situation is far from understood. The talk
will describe new infinite series both of existent coverings and of
exceptions, including previously unknown exceptions with the covering
surface not the sphere and with more than three branching points. All
these series come with systematic explanations, based on three different
techniques (dessins d'enfants, decomposability, graphs on surfaces) that
Pervova and the speaker have exploited to attack the problem, besides
Hurwitz's classical technique based on permutations.
Wednesday April 5, 2006 at 3:00 PM in SEO 427