2015/11/09 by Charalampos Stylianakis, Stylianakis, Charalampos · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Representation Theory (math.RT) #math.GR #math.GT #math.RT
paper · pdf · doi:10.48550/arxiv.1511.02912
22 pages, 8 figures
arxiv created 2015/11/09 · openalex publication_date 2015/11/09 · arxiv updated 2015/11/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.