Study of an identity

We solve the word problem of the identity x(yz) = (xy)(yz) by investigating a certain group describing the geometry of that identity. We also construct a concrete realization of the one-generated free algebra relative to the above identity. Received March 23, 2001; accepted in final form July 6, 2002.     

Combinatorics of normal sequences of braids

Many natural counting problems arise in connection with the normal form of braids—and seem to have not been much considered so far. Here we solve some of them. One of the noteworthy points is that a number of different induction schemes appear. The key technical ingredient is an analysis of the normality condition in terms of permutations and their descents, in the vein of the Solomon algebra. As was perfectly summarized by a referee, the main result asserts that the size of the automaton involved in the automatic structure of Bn associated with the normal form can be lowered from n! to p(n), the number of partitions of n.     

Free zeropotent left distributive systems

We prove a result conjectured by J.Je[zcirc]ek in [9],namely that a zeropotent left self-distributive system need not be 3-trivial, i.e.,the identity x(yz)=0 does not follow from the identities x(yz)=(xy)(xz),xx=0 and x0=0x=0.The argument is a general scheme possibly working for various identities in the context of self-distributivity.     

On the rotation distance between binary trees

We develop combinatorial methods for establishing lower bounds on the rotation distance between binary trees, i.e., equivalently, on the flip distance between triangulations of a polygon. These methods lead to sharp estimates for certain particular pairs of trees. As an application, we prove that, for each n, there exist size n trees at distance , i.e., the diameter of the nth associahedron has at least this value.     

Three-Dimensional Realizations of Braids

Consider a standard braid diagram as a three-dimensional figureviewed from the top; what happens when this figure is lookedat from the side? Then a new braid can be obtained, and studyingthe connection between the initial braid and the derived braidso obtained provides both a new simple proof for the existenceof the right greedy normal form of positive braids and a geometricalinterpretation for the automatic structure of the braid groups.     

The group of parenthesized braids

We investigate a group B_• that includes Artin's braid group B_∞ and Thompson's group F. The elements of B_• are represented by braids diagrams in which the distances between the strands are not uniform and, besides the usual crossing generators, new rescaling operators shrink or stretch the distances between the strands. We prove that B_• is a group of fractions, that it is orderable, admits a nontrivial self-distributive structure, i.e., one involving the law x(yz)=(xy)(xz), it embeds in the mapping class group of a sphere with a Cantor set of punctures, and that Artin's representation of B_∞ into the automorphisms of a free group extends to B_•.     

Almost commensurability of 3-dimensional Anosov flows

Two flows are almost commensurable if, up to removing finitely many periodic orbits and taking finite coverings, they are topologically equivalent. We prove that all suspensions of automorphisms of the 2-dimensional torus and all geodesic flows on unit tangent bundles to hyperbolic 2-orbifolds are pairwise almost commensurable.