Title:

Some applications of geometric techniques in combinatorial group theory

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 starcomplexes of
complexes. Next we present background material on diagrams and
pictures. The final section in the chapter deals with
SQuniversality. 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 nontrivial 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
propertyWk if no nontrivial 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 propertyW2
(II) r is trianglefree and each Ge has propertyWI'
Assuming that (I) or (II) holds we: (i) prove a
Freihietssatz for these groups; (ii) give sufficient
conditions for the groups to be SQuniversal; (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 starcomplexes and
coverings to look at NECgroups.
An NEC (NonEuclidean Crystallographic) group is a
discontinuous group of isometries (some of which may be
(vi)
orientation reversing) of the NonEuclidean plane. According
to Yilkie [46], a finitely generated NECgroup with compact
orbit space has a presentation as follows:
Involutary generators: Yij (i,j)EZo
Noninvolutary 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 NEecomplex. (This involves a
structural re$triction on the form of the starcomplex of the
complex.) It is obvious from the definition that this class of
complexes is clo$ed under coverings, so that the class of
fundamental groups of NEecomplexes is trivially closed under
taking subgroups. We then obtain structure theorems for both
finite and infinite NEecomplexes.
We show that the fundamental group of a finite NEecomplex
has a presentation of the form (*) and that the fundamental
group of an infinite NEecomplex 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 NEegroups given in [5] and [6].
,
(viii)
In chapter 4 we use the techniques of coverings and
diagrams. to stue,iy the SQuniversau'ty of Coxeter groups. This
is a problem due to B.H. Neumann (unpublished). see [40].
A Coxeter pair is a 2tup1e (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 SQuniversa1. 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 SQuniversal.
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 SQuniversal 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 SQuniversal,
does this imply that one of the summands is itself
SQuniversal?
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
