Galois action on homology of generalized Fermat Curves

The fundamental group of Fermat and generalized Fermat curves is computed. These curves are Galois ramified covers of the projective line with abelian Galois groups H. We provide a unified study of the action of both cover Galois group H and the absolute Galois group $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ on the pro-$\ell$ homology of the curves in study. Also the relation to the pro-$\ell$ Burau representation is investigated.

Classification of the real discrete specialisations of the Burau representation of B3

This classification is found by analyzing the action of a normal subgroup of B3 as hyperbolic isometries. This paper gives an example of an unfaithful specialisation of the Burau representation on B4 that is faithful when restricted to B3, as well as examples of unfaithful specialisations of B3.

Burau representation for n = 4

The problem of faithfulness of the (reduced) Burau representation for [Formula: see text] is known to be equivalent to the problem of whether certain two matrices [Formula: see text] and [Formula: see text] generate a free group of rank two. In this note, we give a simple proof that [Formula: see text] is a free group of rank two.

Burau Maps and Twisted Alexander Polynomials

The Burau representation of the braid group can be used to recover the Alexander polynomial of the closure of a braid. We define twisted Burau maps and use them to compute twisted Alexander polynomials.

The level four braid group

By evaluating the Burau representation att=-1, one obtains a symplectic representation of the braid group. We study the resulting congruence subgroups of the braid group, namely, the preimages of the principal congruence subgroups of the symplectic group. Our main result is that the level four congruence subgroup is equal to the group generated by squares of Dehn twists. We also show that the image of the Brunnian subgroup of the braid group under the symplectic representation is the level four congruence subgroup.