It is known that a number of algebraic properties of the braid groups extend to arbitrary finite Coxeter type Artin groups. Here we show how to extend the results to more general groups that we callâ€¦ (More)

Define a Garside monoid to be a cancellative monoid where right and left lcmâ€™s exist and that satisfy additional finiteness assumptions, and a Garside group to be the group of fractions of a Garsideâ€¦ (More)

We survey some of the recently developed cryptographic schemes involving Artinâ€™s braid groups, as well as the attacks against these schemes. We also point out some hints for future work.

Conjugacy is not the only possible primitive for designing braidbased protocols. To illustrate this principle, we describe a Fiatâ€“Shamir-style authentication protocol that be can be implemented usingâ€¦ (More)

Starting from a certain monoid that describes the geometry of the left self-distributivity identity, we construct an explicit realization of the free left self-distributive system on any number ofâ€¦ (More)

â€” In the decade since the discovery that Artinâ€™s braid groups enjoy a left-invariant linear ordering, several quite different approaches have been applied to understand this phenomenon. This book isâ€¦ (More)

We investigate a group Bâ€¢ that includes Artinâ€™s braid group Bâˆž and Thompsonâ€™s group F . The elements of Bâ€¢ are represented by braids diagrams in which the distances between the strands are notâ€¦ (More)

We describe a new method for comparing braid words which relies both on the automatic structure of the braid groups and on the existence of a linear ordering on braids. This syntactical algorithm isâ€¦ (More)

An extension of the infinite braid group Bâˆž exactly describes the left distributivity identities. These identities form a decidable family. The group Bâˆž is closely connected with the free leftâ€¦ (More)