We review properties of reductive groups related to existence of a (B, N)-pair. For an abstract group, having a (B, N)-pair is a very strong condition; many of the theorems we will give for reductive groups follow from this single property. Definition 3.1.1 We say that two subgroups B and N of a group G form a (B, N)-pair (also called a Tits system) for G if: (i) B and N generate G and T := B ∩ N is normal in N. (ii) The group W := N/T is generated by a set S of involutions such that: (a) For s ∈ S, w ∈ W we have BsB.BwB ⊂ BwB ∪ BswB.

This text consists of the introduction, table of contents, and bibliography of a long manuscript (703 pages) that is currently submitted for publication. This manuscript develops an extension of Garside's approach to braid groups and provides a unified treatment for the various algebraic structures that appear in this context. The complete text can be found at http://www.math.unicaen.fr/∼garside/Garside.pdf. Comments are welcome.

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We state a conjecture about centralizers of certain roots of central elements in braid groups, and check it for braid groups of type A, B, G (d, 1, r) and a couple of other cases. Our proof makes use of results from Birman-Ko-Lee, of which we give a new intrinsic account.