In this paper we study the reduction curves of a braid, and how they can be used to decompose the braid into simpler ones in a precise way, which does not correspond exactly to the decomposition given by Thurston theory. Then we study how a cyclic sliding (which is a particular kind of conjugation) affects the normal form of a braid with respect to the normal forms of its components. Finally, using the above methods, we provide the example of a family of braids whose sets of sliding circuits (hence ultra summit sets) have exponential size with respect to the number of strands and also with respect to the canonical length.

An embedding of the m-times punctured disc into the n-times punctured disc, for n > m, yields an embedding of the braid group on m strands Bm into the braid group on n strands Bn, called a geometric embedding. The main example consists of adding n−m trivial strands to the right of each braid on m strands. We show that geometric embeddings do not merge conjugacy classes, meaning that if the images of two elements in Bm by a geometric embedding are conjugate in Bn, the original elements are conjugate in Bm. We also show that the result does not hold, in general, for geometric embeddings of mapping class groups.

Geometric approaches to braid groups and mapping class groups AbstractThese are Lecture Notes of a course given by the author at the School Winter Braids, held at the Université de Pau et des Pays de L'Adour (France), on February 2015. It is explained how mapping class groups, and in particular braid groups, act on some interesting geometric spaces like the hyperbolic plane and the complex of curves, and how this allows to obtain some algebraic properties of the groups. A proof of the hyperbolicity of the graph of curves, following Hensel-Przytycki-Webb, is given.