Quilts: central extensions, braid actions, and finite groups
type: Item
instance of: scholarly article
title: Quilts: central extensions, braid actions, and finite groups
zbMATH Open document ID: 0998.20035
author: Tim Hsu
published in: Lecture Notes in Mathematics
publication date: 2000-06-26T00:00:00Z
review text: This monograph has two main parts. Part I (Chaps. 3-7) concentrates on the basic definitions and fundamental results on quilts and Norton systems, and Part II (Chaps. 8-13) addresses the structure problem, especially as it relates to finite groups. In Chapter 2 the author reviews some background material. Of particular note are the sections on triangle groups (Sect. 2.2), Seifert groups (Sect. 2.3), and braid groups (Sect. 2.4) since they contain non-standard notation used frequently in the rest of the monograph. In Chapter 3 modular quilts are defined to be certain 2-complexes. Modular quilts classify subgroups of triangle groups (or alternately, transitive permutation representations of triangle groups). Quilts are then defined to be certain equivalence classes of annotated modular quilts, and it is shown that quilts classify subgroups of Seifert groups (or alternately, transitive permutation representations of Seifert groups). The main new idea is that, since Seifert groups are central extensions of triangle groups, any central information that we loose by considering a subgroup of a Seifert group as a subgroup of a triangle group may be regained by a suitable annotation. In Chapter 4, Norton systems are defined to be the orbits of Norton's action of \({\mathbf B}_3\) on pairs of elements of a group, and the author shows that the rules for making the quilt of a Norton system from Sect. 1.3 hold rigorously, albeit with a few necessary corrections. The author then provides some relatively small but useful examples of quilts in Chapter 5, concentrating on quilts of Norton systems. The author then proceeds in Chapters 6 and 7 to his main results on the structure of quilts (not necessarily of Norton systems). Specifically, in Chapter 6, using elementary homology theory, the subgroups of Seifert groups that project down to a given subgroup of the corresponding triangle groups are completely classified. In Chapter 7 the author interprets the results of Chapter 6 in term…
reviewed by: Mina Teicher
Mathematics Subject Classification ID: 20F36
Mathematics Subject Classification ID: 20D08
Mathematics Subject Classification ID: 20-02
Mathematics Subject Classification ID: 20E22
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 1465090
zbMATH Keywords: triangle groups
zbMATH Keywords: Seifert groups
zbMATH Keywords: braid groups
zbMATH Keywords: modular quilts
zbMATH Keywords: subgroups of triangle groups
zbMATH Keywords: central extensions
zbMATH Keywords: Norton systems
zbMATH Keywords: involutions
zbMATH Keywords: Monster
zbMATH Keywords: generators
zbMATH Keywords: presentations
zbMATH Keywords: monodromy actions
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1465090