Given the fundamental group Γ of a surface, we denote by Γ(n) its quotients by the n-th power of every simple closed curve. Such a group appears naturally when studying quantum representations or the quotients of the mapping class group by large powers of Dehn twists.

Nevertheless the nature of Γ(n) as a group is, on the face of it, mysterious: it initially appears to occupy an intermediate status between a surface group and a Burnside group, and might appear to be closer to a Burnside group. In this talk we will present some results suggesting that it is in fact closer to a surface group (e.g. it is virtually torsion free, it fits in a natural Birman-type short exact sequence, etc).

Joint work with A. Sisto and H. Wilton