5,908 research outputs found
Finding involutions with small support
We show that the proportion of permutations in or such that
has even order and is an involution with support of cardinality
at most is at least a constant multiple of
. Using this result, we obtain the same conclusion for elements in
a classical group of natural dimension in odd characteristic that have even
order and power up to an involution with -eigenspace of dimension at most
for a linear or unitary group, or for a symplectic or orthogonal group
Finding involutions with small support
We show that the proportion of permutations in or such that
has even order and is an involution with support of cardinality
at most is at least a constant multiple of
. Using this result, we obtain the same conclusion for elements in
a classical group of natural dimension in odd characteristic that have even
order and power up to an involution with -eigenspace of dimension at most
for a linear or unitary group, or for a symplectic or orthogonal group
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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