On Presentations of Surface Braid Groups

We give presentations of braid groups and pure braid groups on surfaces and we show some properties of surface pure braid groups. 1. Presentations for surface braids Let F be an orientable surface and let P = {P1, . . . , Pn} be a set of n distinct points of F . A geometric braid on F based at P is an n-tuple Ψ = (ψ1, . . . , ψn) of paths ψi : [0, 1] → F such that • ψi(0) = Pi, i = 1 . . . , n; • ψi(1) ∈ P , i = 1 . . . , n; • ψ1(t), . . . , ψn(t) are distinct points of F for all t ∈ [0, 1]. The usual product of paths defines a group structure on the set of braids up to homotopies among braids. This group, denoted B(n, F ), does not depend on the choice of P and it is called the braid group on n strings on F . On the other hand, let be FnF = F n \∆, where ∆ is the big diagonal, i.e. the n-tuples x = (x1, . . . xn) for which xi = xj for some i 6= j. There is a natural action of Σn on FnF by permuting coordinates. We call the orbit space F̂nF = FnF/Σn configuration space. Then the braid group B(n, F ) is isomorphic to π1(F̂nF ). We recall that the pure braid group P (n, F ) on n strings on F is the kernel of the natural projection of B(n, F ) in the permutation group Σn. This group is isomorphic to π1(FnF ). The first aim of this article is to give (new) presentations for braid groups on orientable surfaces. A p-punctured surface of genus g ≥ 1 is the surface obtained by deleting p points on a closed surface of genus g ≥ 1. Theorem 1.1. Let F be an orientable p-punctured surface of genus g ≥ 1. The group B(n, F ) admits the following presentation (see also section 2.2): • Generators: σ1, . . . , σn−1, a1, . . . , ag, b1, . . . , bg, z1, . . . , zp−1 . • Relations: – Braid relations, i.e. σiσi+1σi = σi+1σiσi+1 ; σiσj = σjσi for |i− j| ≥ 2 .