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

TL;DR

This paper proves that the normal closure of powers of half-twists in the mapping class group of a punctured sphere has infinite index, and explores the structure of related quotients, using Jones representations and Hecke algebra actions.

Contribution

It establishes the infinite index of normal closures of powers of half-twists and analyzes the structure of resulting quotients, introducing a novel reformulation via Jones representations.

Findings

Normal closure of a power of a half-twist has infinite index in the mapping class group.

Quotients of the mapping class group contain free abelian subgroups.

In some cases, quotients contain free nonabelian subgroups.

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.