GEOMETRIC PRESENTATIONS FOR THE PURE BRAID GROUP

Geometric Presentations for the Pure Braid Group

We give several new positive finite presentations for the pure braid group that are easy to remember and simple in form. All of our presentations involve a metric on the punctured disc so that the punctures are arranged “convexly”, which is why we describe them as geometric presentations. Motivated by a presentation for the full braid group that we call the “rotation presentation”, we introduce...

Derivative Varieties and the Pure Braid Group

The Chen Groups of the Pure Braid Group

A b st r ac t . The Chen groups of a group are the lower central series quotients of its maximal metabelian quotient. We show that the Chen groups of the pure braid group Pn are free abelian, and we compute their ranks. The computation of these Chen groups reduces to the computation of the Hilbert series of a certain graded module over a polynomial ring, and the latter is carried out by means o...

The Chen Groups of the Pure Braid Group DANIEL

The Chen groups of a group are the lower central series quotients of its maximal metabelian quotient. We show that the Chen groups of the pure braid group Pn are free abelian, and we compute their ranks. The computation of these Chen groups reduces to the computation of the Hilbert series of a certain graded module over a polynomial ring, and the latter is carried out by means of a Groebner bas...

Two-generator Subgroups of the Pure Braid Group

We show that any two elements of the pure braid group either commute or generate a free group, settling a question of Luis Paris. Our proof involves the theory of 3-manifolds and the theory of group actions on trees.