# Sur les monoïdes des classes de groupes de tresses

Abstract : Hurwitz showed that a branched cover f:M→N of surfaces with branch locus P⊂N determines and is determined, up to inner automorphism of the symmetric group S_m, by a homomorphism π_1(NP, ∗) → S_m . This result reduces the questions of existence and uniqueness of branched covers to combinatorial problems. For a suitable set of generators for π_1(NP, ∗), a representation π_1(NP, ∗) → S_m determines and is determined by a sequence (a_1 , b_1 , . . . , a_g , b_g , z_1, . . . , z_k ) of elements of S_m satisfying [a_1, b_1 ] · · · [a_g , b_g ]z_1 · · · z_k = 1. Thesequence (a_1 , b_1 , . . . , a_g , b_g , z_1 , . . . , z_k ) of permutations is called a Hurwitz system for f .Therefore, to understand the classes of branched covers one need to study the orbits of Hurwitz systems by suitable actions on S^n_m, n = 2g+k. One of such actions is the simultaneous conjugation that leads to the study of the set of double cosets of symmetric groups.In Chapter 1 we bring an exposition of the recent work of Neretin on the multiplicative structure on the set S_∞S^n_∞/S_∞ .In Chapter 2 we aim at extending Neretin’s results to the group B_∞ of finitely supported braids on infinitely many strands. We prove that B_∞B^n_∞/B_∞ admits such a multiplicative structure and explain how this structure is related to similar constructions in Aut(F_∞ ) and GL(∞). We also define a one-parameter generalization of the usual monoid structure on the set of double cosets of GL(∞) and show that the Burau representation provides a functor between the categories of double cosets of B_∞ and GL(∞).The last chapter is dedicated to the study of homomorphisms π_1(NP, ∗) → G, G a discrete group. We give an exposition of the stable classification of such homomorphisms following the work of Samperton and some new results concerning the number of stabilizations necessary to make them equivalent with respect to Hurwitz moves. We also explore a generalization of the classification of finite branched covers by introducing the braid monodromy for surfaces embedded in codimension 2. Following ideas of Kamada we defined a braid monodromy associated to braided surfaces, which correspond to G = B_∞ and study the spherical functions associated to braid group representations.
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Pablo Gonzalez Pagotto. Sur les monoïdes des classes de groupes de tresses. Topologie algébrique [math.AT]. Université Grenoble Alpes, 2019. Français. ⟨NNT : 2019GREAM049⟩. ⟨tel-02524192⟩

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