Quilts: central extensions, braid actions, and finite groups (Q1567708)

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 terms of more standard versions of central extension theory. (For example, he recovers the result of \textit{J. H. Conway, H. S. M. Coxeter}, and \textit{G. C. Shephard} [Tensor, New Ser. 25, 405-418 (1972; Zbl 0236.20029)] mentioned in Sect. 1.1). Part II of the monograph begins in Chapter 8 with the author's formulation of the structure problem in terms of combinatorial group theory. Roughly speaking, the structure problem asks: To what extent does a Norton system for a group \(G\) determine the structure of \(G\)? To give this problem a more rigorous framework, a concept called `the group of a quilt' is defined, and the author approaches the structure problem by examining groups of quilts. Consequently, the majority of Chapter 8 is devoted to finding various algorithms for presenting the group of a quilt; in particular, the homology results of Chapter 6 are used to find particularly concise presentations. In Chapter 9, the techniques of Chapter 8 are used to enumerate the groups of some small quilts. Next, in Chapter 10, the monodromy actions of \({\mathbf B}_n\) on \(n\)-tuples of group elements are considered, and the author shows that the monodromy action of \({\mathbf B}_3\) on triples of involutions of a group is essentially equivalent to Norton's actions on pairs of elements. In Chapter 11, examples of the monodromy action on triples of 6-transpositions (a class of involutions of \(C\) such that for \(x,y\in C\), the order of \(xy\leq 6\)) in the Monster are given.par In Chapter 12, the author obtains assorted results on the structure problem. The highlights of the chapter include examples of finite quilts with infinite groups. Finally, in Chapter 13, some related recent work is discussed, as well as some open questions and directions for further research. In Appendix A the author discusses the problem of finding independent generators for a subgroup \(\Gamma\) of \(\text{PSL}_2(\mathbb{Z})\), that is, generators such that \(\Gamma\) is the free product of the corresponding cyclic groups. Two solutions are given, one of which relies only on the Reidemeister-Schreier method from combinatorial group theory, and the other which relies on quilts, especially the results of Chapter 8. For completeness, the author also includes a graphical description of Reidemesister-Schreier based on lectures given by Conway.