Groups with a complemented presentation Original Research Article

Let G be a group given by a presentation. We study the decomposition of the elements of G as quotients of “positive” elements (the elements of G that can be expressed without using the inverses of the generators) in the special case when the presentation satisfies some syntactical condition. This approach works in particular for Artinʹs braid groups, and results in a very simple quadratic algorithm for solving their word problem.