5,495 research outputs found
Acylindrical hyperbolicity and Artin-Tits groups of spherical type
We prove that, for any irreducible Artin-Tits group of spherical type ,
the quotient of by its center is acylindrically hyperbolic. This is
achieved by studying the additional length graph associated to the classical
Garside structure on , and constructing a specific element of
whose action on the graph is loxodromic and WPD in the sense of
Bestvina-Fujiwara; following Osin, this implies acylindrical hyperbolicity.
Finally, we prove that "generic" elements of act loxodromically, where the
word "generic" can be understood in either of the two common usages: as a
result of a long random walk or as a random element in a large ball in the
Cayley graph.Comment: Proof in Section 4 has been simplifie
On the complexity of braids
We define a measure of "complexity" of a braid which is natural with respect
to both an algebraic and a geometric point of view. Algebraically, we modify
the standard notion of the length of a braid by introducing generators
, which are Garside-like half-twists involving strings
through , and by counting powered generators as
instead of simply . The geometrical complexity is some
natural measure of the amount of distortion of the times punctured disk
caused by a homeomorphism. Our main result is that the two notions of
complexity are comparable. This gives rise to a new combinatorial model for the
Teichmueller space of an times punctured sphere. We also show how to
recover a braid from its curve diagram in polynomial time. The key r\^ole in
the proofs is played by a technique introduced by Agol, Hass, and Thurston.Comment: Version 2: added section on Teichmueller geometry, removed section on
train track
Fast algorithmic Nielsen-Thurston classification of four-strand braids
We give an algorithm which decides the Nielsen-Thurston type of a given
four-strand braid. The complexity of our algorithm is quadratic with respect to
word length. The proof of its validity is based on a result which states that
for a reducible 4-braid which is as short as possible within its conjugacy
class (short in the sense of Garside), reducing curves surrounding three
punctures must be round or almost round.Comment: One minor error corrected (Example 4.2 was wrong
Lawrence-Krammer-Bigelow representations and dual Garside length of braids
We show that the span of the variable in the Lawrence-Krammer-Bigelow
representation matrix of a braid is equal to the twice of the dual Garside
length of the braid, as was conjectured by Krammer. Our proof is close in
spirit to Bigelow's geometric approach. The key observation is that the dual
Garside length of a braid can be read off a certain labeling of its curve
diagram
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