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«Combinatorial Group Theory Charles F. Miller III 7 March, 2004 Abstract An early version of these notes was prepared for use by the participants in ...»

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Combinatorial Group Theory

Charles F. Miller III

7 March, 2004

Abstract

An early version of these notes was prepared for use by the participants in

the Workshop on Algebra, Geometry and Topology held at the Australian

National University, 22 January to 9 February, 1996. They have subsequently

been updated and expanded many times for use by students in the subject

620-421 Combinatorial Group Theory at the University of Melbourne.

Copyright 1996-2004 by C. F. Miller III.

Contents 1 Preliminaries 3

1.1 About groups........................... 3

1.2 About fundamental groups and covering spaces........ 5 2 Free groups and presentations 11

2.1 Free groups............................ 12

2.2 Presentations by generators and relations............ 16

2.3 Dehn’s fundamental problems.................. 19

2.4 Homomorphisms......................... 20

2.5 Presentations and fundamental groups............. 22

2.6 Tietze transformations...................... 24

2.7 Extraction principles....................... 27 3 Construction of new groups 30

3.1 Direct products.......................... 30

3.2 Free products........................... 32

3.3 Free products with amalgamation................ 36

3.4 HNN extensions.......................... 43

3.5 HNN related to amalgams.................... 48

3.6 Semi-direct products and wreath products........... 50 4 Properties, embeddings and examples 53

4.1 Countable groups embed in 2-generator groups......... 53

4.2 Non-finite presentability of subgroups.............. 56

4.3 Hopfian and residually finite groups............... 58

4.4 Local and poly properties.................... 61

4.5 Finitely presented coherent by cycli

–  –  –

Chapter 1 Preliminaries The reader is assumed to be familiar with the basics of group theory including the isomorphism theorems and the structure theory of finitely generated abelian groups. The groups we will be dealing with are usually infinite and non-abelian. In relation to several topics the reader will also be expected to be familiar with the theory of fundamental groups and covering spaces. In this chapter we establish some notational conventions and collect some frequently used facts from both elementary group theory and from the theory of covering spaces.

1.1 About groups We normally write groups multiplicatively: if G is a group and g, h ∈ G then their product is gh, the identity is 1 ∈ G and the inverse of g is g −1. Two elements g, h ∈ G are said to commute if gh = hg. Observe that gh = hg ↔ h−1 gh = g ↔ g −1 h−1 gh = 1.

For any elements u, v ∈ G we define their commutator [u, v] = u−1 v −1 uv which measures whether or not u and v commute.

The element v −1 uv is called the conjugate of u by v, while vuv −1 is the conjugate of u by v −1. Two elements u, w ∈ G are said to be conjugate in G if there exists and element v ∈ G such that w = v −1 uv. Being conjugate is an equivalence relation. For fixed v ∈ G the function ιv : G → G defined by ιv (g) = v −1 gv is called conjugation by v. It is a homomorphism since ιv (gh) = v −1 ghv = v −1 gvv −1 hv = ιv (g)ιv (h) and in fact it is an automorphism of G (that is, an isomorphism of G with itself) called an inner automorphism.

If S is a subset of G, we denote by S the smallest subgroup of G containing S, called the subgroup generated by S. This subgroup S can be characterized as the set of all g ∈ G which are equal to a (possibly empty) product of si ∈ S and their inverses, that is {g ∈ G | g = s11 s22 · · · skk for some si ∈ S and = ±1}.

i To see this, just observe that the collection of elements having such expressions contains S and is closed under products and also taking inverses since (s11 s22 · · · skk )−1 = s− k · · · s− 2 s− 1.

k Similarly if R is a subset of G we denote by either nmG (R) or sometimes R G the smallest normal subgroup of G containing R, called the normal closure of R in G. This normal subgroup nmG (R) can be characterized as the set of all g ∈ G which are equal to a (possibly empty) product of conjugates of ri ∈ R and their inverses, that is the set of g ∈ G such that

–  –  –

Of course this is just the subgroup generated by all the conjugates of elements of R. To see that it is a normal subgroup we observe that, for g expressed as above,

–  –  –

One particularly important subgroup of any group G is its commutator subgroup [G, G] which is defined to be the subgroup generated by all the commutators [g, h] with g, h ∈ G. Since the conjugate of a commutator is again a commutator, [G, G] is a normal subgroup and the quotient group G/[G, G] is abelian because every commutator belongs to [G, G]. If we suppose φ : G → A where A is abelian, then φ([g, h]) = [φ(g), φ(h)] =A 1 since A is abelian and so [G, G] ⊆ ker φ. Thus G/[G, G] is the largest abelian quotient group of G. The subgroup [G, G] has the additional property that it is fully invariant, meaning that it is mapped into itself by any homomorphism ψ : G → G. (Note that a fully invariant subgroup is necessarily normal.) Another subgroup of interest is the center Z(G) consisting of all elements z such that zg = gz for all g ∈ G. It is easy to see that Z(G) is a normal subgroup of G and moreover that it is characteristic, meaning that it is invariant under any automorphism of G.





The commutator subgroup [G, G] is sometimes called the derived group and denoted G. Next one can define G = [G, G ] = [[G, G], [G, G]] which is the commutator subgroup of G. Inductively one defines a descending series of subgroups by G(n+1) = [G(n), G(n) ] called the derived series of G of which G and G are the first two terms. The G(k) are fully invariant subgroups and the successive quotients G(k) /G(k+1) are abelian. A group G is solvable (or soluble) of derived length ≤ n if G(n) = 1. In this case G can be constructed by taking successive extensions by the abelian groups G(k) /G(k+1). In the particular case that G(2) = G = 1, the derived group G of G is abelian and G is said to be metabelian. As an example, the multiplicative group of invertible n × n upper triangular matrices over any commutative ring is solvable of derived length n − 1.

Another commonly studied descending series of fully invariant subgroups of G is the lower central series defined by γ1 (G) = G and γn+1 = [γn (G), G].

Here, if H, K are subgroups of G, then [H, K] is the sugroup generated by the commutators [h, k] with h ∈ H, k ∈ K. Thus γ2 (G) = [G, G] and γ3 (G) = [[G, G], G]. A group is said to be nilpotent of class c if γc+1 (G) = 1.

So a group G is nilpotent of class 2 if [[G, G], G] = 1 which is equivalent to saying that [G, G] ⊆ Z(G) or “commutators are central in G”. More generally γc+1 (G) = 1 means the γc (G) is contained in the center of G. Some examples: if p is a prime, a finite p-group is always nilpotent (of some class) and a finite nilpotent group is a direct product of its Sylow p-subgroups.

The multiplicative group of n × n upper triangular matrices over Z (or any commutative ring) with 1’s on the diagonal is nilpotent of class n − 1.

1.2 About fundamental groups and covering spaces Useful references for fundamental groups and covering spaces are the textbooks by Massey [6] and Rotman [8]. We here summarize (with almost no proofs) some of the basic facts we will need.

The spaces we will deal with will always be CW-complexes. These can be thought of as cell complexes which are built inductively by attaching the boundaries of new standard n-cells to an existing complex. Thus a CWcomplex X is a Hausdorff space which is the union of an increasing sequence of subspaces X 0 ⊂ X 1 ⊂ X 2 ⊂ · · · ⊆ X. The initial 0-skeleton X 0 is a discrete set of points. The n-skeleton X n is obtained from X n−1 by attaching n-cells along their boundary (see [6] or [8] for details). X is endowed with the weak topology meaning that a subset A of X is closed if its intersection with each n-cell of X is closed. Since we are largely interested in fundamental groups, we will normally need only 2-dimensional CW-complexes, that is X = X 2.

We assume the reader is familiar with the fundamental group. If x0 is a 0-cell of X, the fundamental group π1 (X, x0 ) is the collection of homotopy classes of loops beginning and ending at x0 relative to this endpoint. The group multiplication is composition of paths meaning first go along one and then the other. Any path with endpoints in X 0 is homotopic relative to its endpoints to a path in X 1. An homotopy between two such paths in X1 can be pushed into X 2 which is why we usually only need 2-dimensional complexes.

The main tool for computing the fundmental group is the following.

Theorem 1.1 (Seifert-vanKampen) Let X = U ∪ V where U, V and U ∩ V are all non-empty, arc-wise connected open subsets of X.

Choose a base point x0 ∈ U ∩ V for all of their fundamental groups. Then the diagram of fundamental groups with maps induced by inclusions

–  –  –

is a pushout. That is, if G is a group and α : π1 (U ) → G and β : π1 (V ) → G are homomorphisms such that ασ = βτ then there is a unique homomorphism

–  –  –

Usually there are ways around the requirement that U and V be open and it is enough to assume they are sub-complexes. Here is the special case which is of most interest to us.

Corollary 1.2 Suppose that the CW-complex X is obtained from the connected CW-complex Y attaching a single 2-cell by identifying its boundary with the loop λ ⊆ Y 1 based at x0. Then π1 (X, x0 ) is isomorphic to the quotient group of π1 (Y, x0 ) by the normal closure of the element [λ] ∈ π1 (Y, x0 ).

Proof : Think of the 2-cell as the unit ball B 2 in R2. Let V be the image of the interior of B 2 in X and let U be Y union the image of B 2 minus the origin. Now U ∩ V is V minus the image of the origin and so is topologically an open annulus with π1 (U ∩ V ) ∼ Z. Since V is topologically an open disk = we have π1 (v) = 1. Hence, by the Seifert-vanKampen Theorem, π1 (X) is the quotient of π1 (U ) by the normal closure of a loop corresponding to the generator of π1 (U ∩ V ). But Y is a deformation retract of U and under the P retraction the annular region U ∩ V maps simply onto λ.

We now turn to covering spaces. For convenience we will assume all spaces are arc-wise connected (path connected) and locally arc-wise connected unless otherwise specified. Since we have in mind CW-complexes the local topology is always well behaved for our applications.

Let X be (path connected) topological space. A covering space is a path ˜ ˜ connected space X together with a continuous mapping p : X → X such that for each x ∈ X there is an open neighborhood U = Ux of x that is evenly covered by p, that is, p−1 (U ) is a disjoint union of open sets Si called sheets such that p|Si : Si → U is a homeomorphism for each i. The map p is often called the covering projection. If x0 ∈ X the discrete set of points p−1 (x0 ) is called the fiber (or fibre) over x0.

By a path in a space X we mean a continuous map λ : I → X where I = [0, 1], the unit interval. Some important properties of covering spaces are the following.

–  –  –

This bring us to the existence problem for covering spaces. On needs some local conditions on the topology of the space X. These conditions do hold for X a CW-complex, so we state this vaguely as the following.

–  –  –

This completes our sketch of covering space theory.

Chapter 2 Free groups and presentations In introductory courses on

Abstract

algebra one is likely to encounter the dihedral group D3 consisting of the rigid motions of an equilateral triangle onto itself. The group has order 6 and is conveniently described by giving two generators which correspond to rotation through 120◦ and flipping about a central axis. These operations have orders 3 and 2 respectively and the group D3 is described by the presentation

D3 = a, b | a2 = 1, b3 = 1, a−1 ba = b−1

some equivalent version. Here the symbols a and b are called generators and the equations they are subjected to are called defining relations.

Combinatorial group theory is concerned with groups described by generators and defining relations and also with certain natural constructions for making new groups out of groups we already have in hand. Also one would like to (1) say something about their structure, their subgroups, and various properties they might enjoy and to (2) find algorithms for answering some natural questions about them and their elements. Combinatorial group theory has many connections with algebraic and geometric topology which have provided both motivation and methods for studying groups in this manner.

In order to begin our study of presentations we first need to discuss free groups. We will then introduce presentations in terms of generators and relations more formally and indicate their connection with algebraic topology via the fundamental group.

–  –  –

and multiplication is defined by ai · aj = ai+j for i, j ∈ Z. Then C is a free group with free basis the set with a single element S = {a}. For if ϕ : S → G is any function, say ϕ(a) = g ∈ G then ϕ extends to a homomorphism ϕ : C → G by defining ϕ(ai ) = g i. Moreover it is clear this is the only way ˜ ˜ to extend ϕ to a homomorphism. Notice that C also has another free basis, namely the singleton set {a−1 } and that these two are the only free bases for C.



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