One of the first interesting representations of the braid groups is the Burau representation. It is the first of the family of Lawrence representations, defined topologically by viewing the braid group as the mapping class group of a punctured disc. Famously, the Burau representation is almost never faithful, but the k = 2 Lawrence representation is always faithful: this is a celebrated theorem of Bigelow and Krammer and it implies that braid groups are linear (embed into general linear groups over a field).

Motivated by this, and by the open question of whether mapping class groups are linear, I will describe recent joint work with Christian Blanchet and Awais Shaukat in which we construct analogues of the Lawrence representations for mapping class groups of compact, orientable surfaces. Tools include twisted Borel-Moore homology of configuration spaces, Schrödinger representations of discrete Heisenberg groups and the Stone-von Neumann theorem.

3.10.pdf