The normal closure of a power of a half-twist has infinite index in the mapping class group of a punctured sphere

Abstract: In this paper we show that the normal closure of the mth power of a half-twist has infinite index in the mapping class group of a punctured sphere. Furthermore, in some cases we prove that the quotient of the mapping class group of the punctured sphere by the normal closure of a power of a half-twist contains a free abelian subgroup. As a corollary we prove that the quotient of the hyperelliptic mapping class group of a surface of genus at least two by the normal closure of the mth power of a Dehn twist has infinite order, and for some integers m the quotient contains a free nonabelian subgroup. As a second corollary we recover a result of Coxeter: the normal closure of the mth power of a half-twist in the braid group of at least five strands has infinite index if n is at least four. Our method is to reformulate the Jones representation of the mapping class group of a punctured sphere, using the action of Hecke algebras on W-graphs, as introduced by Kazhdan-Lusztig.