Crystallographic structures and braid theory

Exposé dans le cadre du séminaire d’algèbre et de géométrie

Orateur : Oscar Ocampo (Salvador de Bahia)

Let M be a compact surface without boundary, and n≥2. We analyse the quotient group Bn(M)/Γ2(Pn(M)) of the surface braid group Bn(M) by the commutator subgroup Γ2(Pn(M)) of the pure braid group Pn(M). If M is different from the 2-sphere S2, we prove that Bn(M)/Γ2(Pn(M))≅Pn(M)/Γ2(Pn(M))⋊φSn, and that Bn(M)/Γ2(Pn(M)) is a crystallographic group if and only if M is orientable. 

If M is orientable, we show a number of results regarding the structure of Bn(M)/Γ2(Pn(M)). Finally, we construct a family of Bieberbach subgroups G˜n,g of Bn(M)/Γ2(Pn(M)) of dimension 2ng and whose holonomy group is the finite cyclic group of order n, and if Xn,g is a flat manifold whose fundamental group is G˜n,g, we prove that it is an orientable Kähler manifold that admits Anosov diffeomorphisms. Joint work with Daciberg Lima Gonçalves, John Guaschi and Carolina de Miranda e Pereiro.