6 research outputs found
On the distance between the expressions of a permutation
We prove that the combinatorial distance between any two reduced expressions
of a given permutation of {1, ..., n} in terms of transpositions lies in
O(n^4), a sharp bound. Using a connection with the intersection numbers of
certain curves in van Kampen diagrams, we prove that this bound is sharp, and
give a practical criterion for proving that the derivations provided by the
reversing algorithm of [Dehornoy, JPAA 116 (1997) 115-197] are optimal. We also
show the existence of length l expressions whose reversing requires C l^4
elementary steps
A conjecture about Artin-Tits groups
We conjecture that the word problem of Artin-Tits groups can be solved
without introducing trivial factors ss^{-1} or s^{-1}s. Here we make this
statement precise and explain how it can be seen as a weak form of
hyperbolicity. We prove the conjecture in the case of Artin-Tits groups of type
FC, and we discuss various possible approaches for further extensions, in
particular a syntactic argument that works at least in the right-angled case
On word reversing in braid groups
International audienceIt has been conjectured that in a braid group, or more generally in a Garside group, applying any sequence of monotone equivalences and word reversings can increase the length of a word by at most a linear factor depending on the group presentation only. We give a counter-example to this conjecture, but, on the other hand, we establish length upper bounds for the case when only right reversing is involved. We also state a new conjecture which would, like the above one, imply that the space complexity of the handle reduction algorithm is linear