Grupos de tranças de superfícies finitamente perfuradas e grupos cristalográficos

Abstract

The link between braid groups on surfaces and crystallographic groups has become such an interesting topic. In the last years some advances were found in the studies of this relation, specially in the case of Artin braid groups and braid groups on closed surfaces (orientable or non-orientable). Our thesis work was strongly inspired by the works in [39] and [42], since here we finish the last cases about surfaces, to which we could ask: is there a relation between braid groups on surfaces and crystallographic groups? Here we analyse, with details, the interaction between braid groups on closed surfaces (orientable or non-orientable) with a finite number of points removed and crystallographic groups. Let X be a closed and finitely punctured surface (orientable or non-orientable). We present new results when X is a closed and finitely punctured surface (orientable or non-orientable) that has a link with crystallographic groups. We prove that the quotient group $B_n(X)\P'_n(X)$ is a crystallographic group, we characterize the finite order elements, i. e., we analyse its torsion subgroup and study the conjugacy classes of the finite order elements. When X is a non-orientable closed and finitely punctured surface with genus $g \geq 2$, we calculate a presentation for the braid groups $P_n(X)$ and $B_n(X)$. In the case of $Pn(X)$, we couldn't find any other presentation in the literature.