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Title: Some applications of geometric techniques in combinatorial group theory
Author: Fennessey, Eric James
ISNI:       0000 0001 3459 7819
Awarding Body: University of Glasgow
Current Institution: University of Glasgow
Date of Award: 1989
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Combinatorial group theory abounds with geometrical techniques. In this thesis we apply some of them to three distinct areas. In Chapter 1 we present all of the techniques and background material neccessary to read chapters 2,3,4. We begin by defining complexes with involutary edges and define coverings of these. We then discuss equivalences between complexes and use these in §§1.3 and 1.4 to give a way (the level method) of simplifying complexes and an application of this method (Theorem 1.3). We then discuss star-complexes of complexes. Next we present background material on diagrams and pictures. The final section in the chapter deals with SQ-universality. The.basic discussion of complexes is taken from notes, by Pride, on complexes without involutary edges, and modified by myself to cover complexes with involution. Chapters 2,3, and 4 are presented in the order that the work for them was done. Chapters 2,3, alld 4 are intended (given the material in chapter 1) to be self contained, and (iv) each has a full introduction. In Chapter 2 we use diagrams and pictures to study groups with the following structure. (a) Let r be a graph with vertex set V and edge set E. We assume that no vertex of r is isolated. (b) For each vertex VEV there is a non-trivial group Gv ' (c) For each edge e-{u,v}EE there is a set Se of cyclically reduced elements of Gu*Gv , each of length at least two. We define Ge to be the quotient of Gu*Gv by the normal closure of Se. We let G be the quotient of *Gv by the normal closure of VEV S- USe. For convenience, we write eEE The above is a generalization ofa situation studied by Pride [35], where each Gv was infinite cyclic.' Let e-{u,v} be an edge of r. We will say that Ge has property-Wk if no non-trivial element of Gu*Gv of free product length less than or equal to 2k is in the kernel of the natural epimorphism (v) We will work with one of the following: (I) Each Ge has property-W2 (II) r is triangle-free and each Ge has property-WI' Assuming that (I) or (II) holds we: (i) prove a Freihietssatz for these groups; (ii) give sufficient conditions for the groups to be SQ-universal; (iii) prove a result which allows us to give long exact sequences relating the (co)-homology G to the (co)-homology of the groups The work in Chapter 2 is in some senses the least original. The proofs are extensions of proofs given in [35] and [39] for the case when each Gv is infinite cyclic. However. there are some technical difficulties which we had to overcome. In chapter 3 we use the two ideas of star-complexes and coverings to look at NEC-groups. An NEC (Non-Euclidean Crystallographic) group is a discontinuous group of isometries (some of which may be (vi) orientation reversing) of the Non-Euclidean plane. According to Yilkie [46], a finitely generated NEC-group with compact orbit space has a presentation as follows: Involutary generators: Yij (i,j)EZo Non-involutary generators: 6i (iElf), tk (l~~r) (*) Defining paths: (YijYij+,)mij (iElf, l~j~n(i)-l) where In Hoare, Karrass and Solitar [22] it is shown that a subgroup of finite index in a group with a presentation of the form (*), has itself a presentation of the form (*). In [22] the same authors show that a subgroup of infinite ingex in a group with a presentation of the form (*) is a free product of groups of the following types: (A) Cyclic groups. (vii) (B) Groups with presentations of the form Xl' ... 'Xn involutary. (e) Groups with presentations of the form Xi (iEZ) involutary. We define what we mean by an NEe-complex. (This involves a structural re$triction on the form of the star-complex of the complex.) It is obvious from the definition that this class of complexes is clo$ed under coverings, so that the class of fundamental groups of NEe-complexes is trivially closed under taking subgroups. We then obtain structure theorems for both finite and infinite NEe-complexes. We show that the fundamental group of a finite NEe-complex has a presentation of the form (*) and that the fundamental group of an infinite NEe-complex is a free product of groups of the forms (A). (B) and (e) above. We then use coverings to derive some of the results on normal subgroups of NEe-groups given in [5] and [6]. , (viii) In chapter 4 we use the techniques of coverings and diagrams. to stue,iy the SQ-universau'ty of Coxeter groups. This is a problem due to B.H. Neumann (unpublished). see [40]. A Coxeter pair is a 2-tup1e (r.~) where r is a graph (with vertex set V(r) and edge set E(r» and ~ is a map from E(r) to {2.3.4 •.•• }. We associate with (r.~) the Coxeter group c(r,~) defined by the presentation tr(r,~)-, where each generator is involutary. Following Appel and Schupp [1] we say that a Coxeter pair is of large type if 2/Im~. I conjecture that if (r,~) is of large type with IV(r)I~3 and r not a triangle with all edges mapped to 3 by ~. then C(r,~) is SQ-universa1. In connection with this conjecture we firstly prove (Theorem 4.1), Let (r,~) be a Coxeter pair of large type. Suppose (A) r is incomplete on at least three vertices, or (B) r is complete on at least five vertices and for 1 < - 2 (ix) Then C(r,~) is SQ-universal. Secondly we prove a result (Theorem 4.2) which shows: If (r,~) is a Coxeter pair with IV(r)I~4 and hcf[~(E(r»] > 1, then C(r,~) is either SQ-universal or is soluble of length at most three. Moreover our Theorem allows us to tell the two possibilities apart. The proof of this result leads to consideration of the following question: If a direct sum of groups is SQ-universal, does this imply that one of the summands is itself SQ-universal? We show (in appendix B) that the answer is "yes" for countable direct sums. We consider the results in chapter 4 and its appendix to be the most significant part of this thesis
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics