3 research outputs found
On minimal crossing number braid diagrams and homogeneous braids
We study braid diagrams with a minimal number of crossings. Such braid
diagrams correspond to geodesic words for the braid groups with standard Artin
generators. We prove that a diagram of a homogeneous braid is minimal if and
only if it is homogeneous. We conjecture that monoids of homogeneous braids are
Artin-Tits monoids and prove that monoids of alternating braids are
right-angled Artin monoids. Using this, we give a lower bound on the growth
rate of the braid groups
Thin groups of fractions
A number of properties of spherical Artin groups extend to Garside groups,
defined as the groups of fractions of monoids where least common multiples
exist, there is no nontrivial unit, and some additional finiteness conditions
are satisfied \cite{Dgk}. Here we investigate a wider class of groups of
fractions, called {\it thin}, which are those associated with monoids where
minimal common multiples exist, but they are not necessarily unique. Also, we
allow units in the involved monoids. The main results are that all thin groups
of fractions satisfy a quadratic isoperimetric inequality, and that, under some
additional hypotheses, they admit an automatic structure
Uniform measures on braid monoids and dual braid monoids
We aim at studying the asymptotic properties of typical positive braids,
respectively positive dual braids. Denoting by the uniform distribution
on positive (dual) braids of length , we prove that the sequence
converges to a unique probability measure on infinite positive
(dual) braids. The key point is that the limiting measure has a
Markovian structure which can be described explicitly using the combinatorial
properties of braids encapsulated in the M\"obius polynomial. As a by-product,
we settle a conjecture by Gebhardt and Tawn (J. Algebra, 2014) on the shape of
the Garside normal form of large uniform braids.Comment: 32 pages, 32 references, 6 tables and 8 figure