Birman's conjecture is true for I₂(p)

James East

Abstract

Joan Birman conjectured that the singular braid monoid embed in the integral group ring of the corresponding braid group. This conjecture can be generalised to all Artin groups. In this article we prove that the conjecture holds for one of the infinite families of Artin groups of finite type, namely I₂(p) We also give a positive presentation of the pure Artin group of the same type which we believe to be new. We show that this presentation defines a monoid which belongs to a special subclass of monoids known as chainable monoids which possess many nice properties.

Keywords: Artin group, Singular Artin monoid.