380 research outputs found
Forgetful maps between Deligne-Mostow ball quotients
We study forgetful maps between Deligne-Mostow moduli spaces of weighted
points on P^1, and classify the forgetful maps that extend to a map of
orbifolds between the stable completions. The cases where this happens include
the Livn\'e fibrations and the Mostow/Toledo maps between complex hyperbolic
surfaces. They also include a retraction of a 3-dimensional ball quotient onto
one of its 1-dimensional totally geodesic complex submanifolds
Braided surfaces and their characteristic maps
We show that branched coverings of surfaces of large enough genus arise as
characteristic maps of braided surfaces, thus being -prems. In the reverse
direction we show that any nonabelian surface group has infinitely many finite
simple nonabelian groups quotients with characteristic kernels which do not
contain any simple loops and hence the quotient maps do not factor through free
groups. By a pullback construction, finite dimensional Hermitian
representations of braid groups provide invariants for the braided surfaces. We
show that the strong equivalence classes of braided surfaces are separated by
such invariants if and only if they are profinitely separated.Comment: 20
Forgetful maps between Deligne-Mostow ball quotients
We study forgetful maps between Deligne-Mostow moduli spaces of weighted
points on P^1, and classify the forgetful maps that extend to a map of
orbifolds between the stable completions. The cases where this happens include
the Livn\'e fibrations and the Mostow/Toledo maps between complex hyperbolic
surfaces. They also include a retraction of a 3-dimensional ball quotient onto
one of its 1-dimensional totally geodesic complex submanifolds
Algorithmic aspects of branched coverings
This is the announcement, and the long summary, of a series of articles on
the algorithmic study of Thurston maps. We describe branched coverings of the
sphere in terms of group-theoretical objects called bisets, and develop a
theory of decompositions of bisets.
We introduce a canonical "Levy" decomposition of an arbitrary Thurston map
into homeomorphisms, metrically-expanding maps and maps doubly covered by torus
endomorphisms. The homeomorphisms decompose themselves into finite-order and
pseudo-Anosov maps, and the expanding maps decompose themselves into rational
maps.
As an outcome, we prove that it is decidable when two Thurston maps are
equivalent. We also show that the decompositions above are computable, both in
theory and in practice.Comment: 60-page announcement of 5-part text, to apper in Ann. Fac. Sci.
Toulouse. Minor typos corrected, and major rewrite of section 7.8, which was
studying a different map than claime
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