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

Stylianakis, C. (2019) The normal closure of a power of a half-twist has infinite index in the mapping class group of a punctured sphere. Journal of Topology and Analysis, 11(2), pp. 273-292. (doi: 10.1142/s1793525319500122)

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 if m is at least five. 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 group. 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 four strands has infinite index. 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.

• Item Type: Articles
• Keywords: Geometry and topology, analysis.
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Stylianakis, Mr Charalampos
• Authors: Stylianakis, C.
• College/School: College of Science and Engineering > School of Mathematics and Statistics
• Journal Name: Journal of Topology and Analysis
• Publisher: World Scientific Publishing
• ISSN: 1793-5253
• ISSN (Online): 1793-7167
• Published Online: 29 August 2017