Lower central series of partitioned braids on surfaces

Partitioned braid groups (sometimes called "mixed braid groups") are subgroups of the braid group standing between the pure braid group Pn and the whole braid group Bn. On the one hand, the lower central series of Bn is almost trivial. On the other hand, the lower central series of Pn is a very rich object, encoding finite type invariants of braids. As a consequence, one can expect partitioned braid groups to display a range of intermediate behaviors, and this is indeed what we observe. In this talk, we will explore these different behaviours and give an answer to the first question one can ask about these lower central series: when do they stop? Even this simple question turns out to be a difficult one, especially when one considers its generalization to braids on surfaces. However, we will be able to answer it almost completely, leaving open only some cases of partitioned braids on the projective plane.