Abstract

The Burau representation is one of the most classical ones for braid groups. Its (lack of) faithfulness has attracted quite a lot of attention, but the case of the 4-strand braid group still remains open. I'll review recent and ongoing work with Asilata Bapat and Thomas Haettel, in two directions: failed attempts to prove faithfulness by geometrical means, and more successful attempts to disprove faithfulness for other Artin-Tits classes.