# The lower central and derived series of the braid groups of compact surfaces

## TL;DR

This paper investigates the lower central and derived series of braid groups on compact surfaces, determining conditions for residual nilpotency and solubility, with explicit results for specific surfaces like the torus and Klein bottle.

## Contribution

It provides new explicit calculations and general descriptions of the lower central and derived series for braid groups on various compact surfaces, including non-orientable cases.

## Key findings

- B extsubscript{n}(M) and P extsubscript{n}(M) are residually nilpotent or soluble for specific n
- Explicit lower central series for P extsubscript{2}(K) where K is the Klein bottle
- Determined residual properties for braid groups on non-orientable surfaces

## Abstract

Let M be a compact surface, either orientable or non-orientable. We study the lower central and derived series of the braid and pure braid groups of M in order to determine the values of n for which B\_n(M) and P\_n(M) are residually nilpotent or residually soluble. First, we solve this problem for the case where M is the 2-torus. We then give a general description of these series for an arbitrary semi-direct product that allows us to calculate explicitly the lower central series of P\_2(K), where K is the Klein bottle, and to give an estimate for the derived series of P\_n(K). Finally, if M is a non-orientable compact surface without boundary, we determine the values of n for which B\_n(M) is residually nilpotent or residually soluble in the cases that were not already known in the literature.