The normal closure of powers of twists in the mapping class group, and the Jones representation

The Jones representation of the mapping class group of the punctured sphere is constructed by formulating irreducible linear representations of braid groups that factor through Hecke algebras. In this talk we introduce the Jones representation and we show that the normal closure of the m-th power of a half-twist has infinite index in the mapping class group of a punctured sphere. As a corollary we show that the normal closure of a power of a Dehn twist has infinite index in the hyperelliptic mapping class group of a closed surface of genus at least two.