an introduction to geometric group theory pristina -...

An introduction to geometric group theoryPristina

Matthieu Dussaule

Mars 2017

This is a eight hours course that I gave at the University of Pristina. It was addressed to students withdifferent backgrounds and knowledge (second, third and fourth year). The goal of the class was first to convincestudents that group theory is nice, secondly that groups can be seen as geometric objects and thirdly that it isalways a good idea to find a space on which a group acts to find out information about the group itself.

The first part is about general theory of groups. The second part is about defining group with a combinatorialapproach. The third part is about the geometric theory.

Exercises are really important too. There are some in the text, but I also added my exercise sheets at theend of the notes.

There are a lot a books about geometric group theory. One I chose as a reference is [2]. It is reallypedagogical and offers a lot of nice perspective. There are a lot of examples and a lot of exercises too. Actually,some of my exercises are taken from this book. Another reference is the first part of [4], also translated intoenglish ([5]). In this one, the reader really understand, I think, that actions of groups on geometric spaces (suchas trees) can be very interesting!

Contents

1 Elementary theory of groups 11.1 Groups, subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Group morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Cosets, normal subgroups and quotient groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Group actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Group presentations 72.1 Free groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 A universal property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Presentations: generating sets and relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Cayley graphs 103.1 Graphs and trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Cayley graphs of a group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 Metric structures on graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.4 Equivalence of Cayley graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1 Elementary theory of groups

Before introducing groups as geometric objects, we first give some basic definitions. The word elementary doesnot mean that everything is trivial in this paragraph. It just means that the geometric approach is not yettackled. However, one should find in here everything that is necessary to understand the remainder of thesenotes. Everything is covered in the two first chapters of [3].

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1.1 Groups, subgroups

Definition 1.1.1. A group is a set G together with a binary operation ‹ : G ˆ G Ñ G that satisfies thefollowing:

1. @a, b, c P G, pa ‹ bq ‹ c “ a ‹ pb ‹ cq,

2. De P G,@a P G, a ‹ e “ e ‹ a “ a,

3. @a P G, Da1 P G, a ‹ a1 “ a1 ‹ a “ e.

The first property is called associativity. The element e is called the neutral element. Sometimes we shall referto it as the unity or the identity. It is unique. The element a1 in the third property is called the inverse of a. Itis also unique. It shall be denoted by a´1. The group is called abelian if for every a and b in G, a ‹ b “ b ‹ a.

From now on, we shall denote the binary operation ‹ with a multiplication symbol. For example, a ‹ b shallbe denoted ab. When the group is abelian, it is standard to use the symbol `.

Let’s give some examples!

1. The set of real numbers R together with the addition ` is a group. The neutral element is 0 and theinverse of x is ´x.

2. The set of non-zero real numbers R˚ together with the multiplication ˆ is a group. The neutral elementis 1 and the inverse of x is 1x.

3. The subset R˚` of R˚ that consists of positive real numbers is also a group together with multiplication.It has the same neutral element and the inverse of x is still 1x.

4. The set of integers Z together with addition is a group. The neutral element is 0 and the inverse of x is´x.

5. Let n be a positive integer and let Sn be the set of bijective maps from t1, ..., nu onto t1, ..., nu. Togetherwith the composition of functions, it is a group.

6. More generally, if X is any set, the set BijpXq of bijective maps from X onto X is a group together withthe composition of functions.

7. Let n be a positive integer and let GLnpRq be the set of invertible matrices of size n. Together with themultiplication of matrices, it is a group.

Definition 1.1.2. If G is a finite group, then its cardinal is called its order.

For example the group Sn is of order n!.

Definition 1.1.3. Let G be a group and H be a subset of G. We say that H is a subgroup of G if

1. e P H,

2. @a, b P H, ab P H,

3. @a P H, a´1 P H.

If H is a subgroup of G, then H together with the operation of G is itself a group. The fact that H is asubgroup of G shall be denoted by H ă G. For example, pZ,`q ă pR,`q and pR˚`,ˆq ă pR˚,ˆq.

Definition 1.1.4. If X is a subset of a group G, the smallest subgroup of G containing X is called thesubgroup generated by X. It is denoted by xXy.

Proposition 1.1.5. The subgroup generated by X always exists.

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Proof. Take xXy as the intersection of all subgroups of G containing X. Actually, check that the intersectionof a family of subgroups is a subgroup.

A generating set for a group G is a set X such that xXy “ G.

Definition 1.1.6. The order of an element x P G is the order of xxy.

Definition 1.1.7. A group is said to be cyclic if it is generated by one single element, meaning that thereexists x P G such that xxy “ G.

If G is cyclic generated by x and if G is finite, then the order of x is the order of G.

1.2 Group morphisms

Definition 1.2.1. Let G and G1 be two groups. A group morphism from G to G1 is a map f : G Ñ G1 suchthat for every a, b P G, fpabq “ fpaqfpbq. The image of f is the set Impfq :“ tx P G1, Dy P G, fpyq “ xu. Thekernel of f is the set Kerpfq :“ tx P G, fpxq “ eu.

Proposition 1.2.2. One always has Impfq ă G1, Kerpfq ă G.

This proposition is really easy to prove and the reader should do it as an exercise. Actually, it is worthnoticing that Impfq ă G1. It is not always the case in some theory that the image of a morphism (when it isdefined) is a sub-object. To give full meaning to this remark, one should talk about category theory. We won’t.

Proposition 1.2.3. A group morphism f is one-to-one if and only if Kerpfq “ teu.

Let’s give examples!

1. The determinant det : GLnpRq Ñ R˚ is a group morphism.

2. The exponential map exp : RÑ R˚` is a group morphism.

3. The logarithm map ln : R˚` Ñ R is a group morphism.

Proposition 1.2.4. If f is a bijective group morphism, then its inverse is also a group morphism.

Proof. Let f : G Ñ G1 be a bijective group morphism and let g : G1 Ñ G be the inverse of f . Let a, b P G1.Then there are x, y P G such that fpxq “ a, fpyq “ b, so that gpabq “ gpfpxqfpyqq “ gpfpxyqq since f is agroup morphism. Thus, gpabq “ xy “ gpaqgpbq.

Definition 1.2.5. A group isomorphism is a group morphism f : GÑ G1 that is bijective. If G1 “ G, thenf is called a group automorphism.

If there exists an isomorphism f : GÑ G1, we shall say that G and G1 are isomorphic and we shall denotethis fact by G » G1.

1.3 Cosets, normal subgroups and quotient groups

Let G be a group and H be a subgroup of G. Say that x P G is equivalent to y P G if there exists h P H suchthat x “ yh. This defines an equivalence relation on G. The equivalence classes are the sets xH “ txh, h P Hu.They are called left cosets.

Remark 1.3.1. One can also define right cosets by saying that x „ y if x “ hy. The equivalence classes arethen the sets Hx.

The quotient of G by this equivalence relation is denoted by GH. It is the set txH, x P Gu.

Remark 1.3.2. For right cosets, one should denote the quotient by HzG.

The quotient map x P G ÞÑ xH P GH is denoted by πH , or just π.

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Definition 1.3.3. The index of H in G is the cardinal of GH. It is denoted by rG : Hs.

Example 1.3.4. Let pZ,`q be the group of integers. For n P Z, let nZ be the set tnk, k P Zu. Then, nZ is asubgroup of Z (actually, every subgroup of Z is of that form). The quotient ZnZ is called the set of integersmodulo n. It is of cardinal |n|. In other words, nZ is of index |n| in Z.

Proposition 1.3.5. If G is finite, then all the cosets xH have the same cardinal.

Proof. Let x P G. Then, the map h P H ÞÑ xh P xH is bijective. In particular, the cardinal of xH is thecardinal of H.

Corollary 1.3.6. If G is finite, then CardpGq “ CardpHqrG : Hs. In particular, the order of H divides theorder of G.

This corollary is often referred as the Lagrange theorem.

Corollary 1.3.7. If G is finite and if x P G, then the order of x divides the order of G.

Definition 1.3.8. If G is a group and H is a subgroup of G, then H is said to be normal in G if for everyx P G, xH “ Hx or equivalently, if xHx´1 “ H.

The fact that H is a subgroup of G shall be denoted by H Ÿ G.

Exercise 1.3.9. 1. If G is abelian, then all is subgroups are normal.

2. Let SLnpRq be the subgroup of GLnpRq that consists of invertible matrices of determinant 1. ThenSLnpRq Ÿ GLnpRq.

3. More generally, if f : GÑ G1 is a group morphism, then Kerpfq Ÿ G.

Theorem 1.3.10. Let G be a group and H be a subgroup of G. There exists a group structure on GH suchthat the quotient map πH is a group morphism if and only if H is normal in G. In that case, the group structureon GH is unique and the kernel of πH is H.

Proof. If such a group structure does exist, then the neutral element of GH is πHpeq “ eH “ H. SinceaH “ H if and only if a P H, one has KerpπHq “ H. Thus, H is normal in G (using the exercise above).Furthermore, aHbH “ πHpaqπHpbq “ πHpabq “ abH so that the group structure is unique.

Conversely, if H is normal, define aHbH “ abH. This definition does not depend on the choices of a and b.Indeed, if aH “ a1H and bH “ b1H, then abH “ aHb “ a1Hb “ a1bH “ a1b1H. This operation is associative,because of associativity in G and aHH “ aH “ HaH, so that H is neutral. Finally, aHa´1H “ aa´1H “ Hand the same works with a´1HaH, so that every coset aH has an inverse, namely a´1H. Therefore, GHtogether with this operation is a group and for that structure, πH is a group morphism.

Remark 1.3.11. Every normal subgroup is then the kernel of some group morphism.

The following theorem is called the first isomorphism theorem (see chapter 2 of [3] for the second and thethird ones). We shall use it a lot in the following.

Theorem 1.3.12. Let G,G1 be two groups and H Ÿ G. Let f : GÑ G1 be a group morphism. There exists agroup morphism f : GH Ñ G1 such that f “ f ˝ πH if and only if H ă Kerpfq. In that case, f is unique.

This property is often summarised with the following diagram.

G

πH

GH

fG1

f

4

Proof. If f does exist, then fpxHq “ fpxq so that is is entirely determined by f . Therefore, it is unique.Furthermore, if h P H, then fphHq “ fphq “ fpHq “ fpeq “ e1, so that H ă Kerpfq.

Conversely, if H ă Kerpfq, define fpxHq “ fpxq. It does not depend on the choice of x. Indeed, ifxH “ x1H, then x “ x1h for some h P H and thus fpxq “ fpx1qfphq “ fpx1q.

Corollary 1.3.13. Let f : GÑ G1 be a group morphism, then GKerpfq » Impfq.

For example, GLnpRqSLnpRq » R˚.Remark 1.3.14. One has now enough material to ask oneself some easily formulated questions with non-trivialanswers. As an example, if H Ÿ G and if GH » t0u, then G “ H. Conversely, if GH » G, is H trivial,meaning H “ t0u ? In general, no! Take G to be the circle S1 and H to be the subgroup t1,´1u of G. ThenGH is isomorphic to S1. Is the answer true if G is finitely generated, meaning that there exists some finite setS such that xSy “ G? The answer still is no. An example is the Baumslag-Solitar group BSp1, 2q.

Actually, those groups such that for every H Ÿ G such that GH » G, H “ t0u have a name, they arecalled Hopfian groups It is equivalent to assume that every group morphism f : G Ñ G that is onto is alsoone-to-one. In fact, Gilbert Baumslag and Donald Solitar considered the group BSp1, 2q to show that therewere non-Hopfian finitely generated groups.

1.4 Group actions

The geometric approach to group theory is all about group actions on geometric spaces. Let’s give somevocabulary and basic properties about those.

Definition 1.4.1. Let G be a group and X be a set. A group action of G on X is a map f : GˆX Ñ X suchthat

1. @x P X, fpe, xq “ x,

2. @g, h P G,@x P X, fpg, fph, xqq “ fpgh, xq.

We shall denote the image of pg, xq by g ¨ x. The properties are then

1. @x P X, e ¨ x “ x,

2. @g, h P G,@x P X, g ¨ ph ¨ xq “ pghq ¨ x.

The fact that G acts on X shall be denoted by G ñ X.If g P G, then the map x P X ÞÑ g ¨ x P X is a bijection of X. Indeed, one has an explicit inverse given by

x ÞÑ g´1 ¨ x. Denote this map by φpgq. Then g ÞÑ φpgq is a group morphism from G to BijpXq.

Proof. For x P X and g, h P G, φpghqpxq “ pghq ¨x “ g ¨ ph ¨xq “ φpgqpφphqpxqq, so that φpghq “ φpgq˝φphq.

Conversely, if φ : GÑ BijpXq is a group morphism, define g ¨ x “ φpgqpxq. It is a group action of G on X.Thus, a group action is nothing else that a group morphism from G to some group BijpXq.

Remark 1.4.2. We shall see later that if X has geometric properties (for example X is a metric space), we shallrequire that the action preserves those properties, meaning that the group morphism GÑ BijpXq has its imagein some subgroup of BijpXq (for example the group of isometries of X).

Definition 1.4.3. A group action G ñ X is called faithful if the morphism G Ñ BijpXq is one-to-one. Inother words, if g P G and if for every x, g ¨ x “ x, then g “ e.

Everything in this notes will be up-to isomorphism. Then, if one has a faithful action G ñ X, one can seeG as a subgroup of BijpXq.

Definition 1.4.4. A group action G ñ X is called free if for every g ‰ e, for every x P X, g ¨ x ‰ x. Inother words, every element different from the neutral element moves every point. A free action is in particularfaithful.

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Let’s give some examples!

1. If G is a group, then G acts on itself by left translation: g ¨ h “ gh. This action is free and hence, it isfaithful. Therefore we can see G a subgroup of BijpGq.

Corollary 1.4.5. If G is finite, then there exists n such that G ă Sn.

Proof. Let n be the order of G, so that BijpGq » Sn. Since G acts faithfully on itself, G ă Sn.

2. A group G also acts on itself by conjugacy: g ¨ h “ g´1hg. This action is not faithful in general. Forexample, if G is abelian, the action is trivial.

3. If H Ÿ G, then G also acts by left translation on GH: g ¨ xH “ gxH.

4. The group GLnpRq acts linearly on Rn. It also acts on the set PpRnq of straight lines of Rn (that goesthrough 0). More generally, let Gn,k be the set of k-dimensional vector-subspaces of Rn. Then GLnpRnqacts on Gn,k.

Definition 1.4.6. Let G be a group, X a set, and assume that G acts on X. If x P X, the orbit of x is theset ωpxq :“ tg ¨ x, g P Gu. The stabiliser of x is the set Stabpxq :“ tg P G, g ¨ x “ xu.

The orbit of x is a subset of X, the stabiliser of x is a subset of G. Actually, the stabiliser of x is a subgroupof G.

Proof. By definition, e¨x “ x. If g, h P Stabpxq, then gh¨x “ g¨ph¨xq “ g¨x “ x and g´1 ¨x “ g´1 ¨pg¨xq “ x.

If G acts on X, say that x, y P X are equivalent if there exists g P G such that g ¨x “ y. It is an equivalencerelation. The equivalence classes are the orbits.

The following is a generalisation of proposition 1.3.5. The map gStabpxq P GStabpxq ÞÑ g ¨ x P ωpxq is welldefined and is a bijection. Then, the cardinal of ωpxq is the index of Stabpxq in G.

In particular, if X is finite, then ωpxq is finite and therefore, Stabpxq is of finite index in G. If G is alsofinite, then CardpGq “ CardpωpxqqCardpStabpxqq.

Theorem 1.4.7 (The class formula). If X is finite, then denote by Ω the set of all orbits and choose one x ineach ω P Ω. Denote this set of such x by X. Then,

CardpXq “ÿ

xPX

rG : Stabpxqs.

This theorem can be really useful. It establishes a link between the knowledge of G and its subgroupsand the knowledge of X. It often allows one to understand G via its action on X. The exercise below is anillustration of this. Actually, in the following, one should keep in mind this guidind principle: Understanding agroup G is understanding how it acts.

Exercise 1.4.8. Let G be a finite group and let p be a prime number such that p divides |G|. We are going toprove that there exists in G some element g of order exactly p: gp “ e and gk ‰ e, if k ď p´ 1.

1. Denote by X the set tpg1, ..., gpq P Gp, g1...gp “ eu. Show that |X| “ |G|p´1. Thus, p divides |X|.

2. Show that j ¨ pg1, ..., gpq :“ pg1`j , ..., gp`jq defines a group action of ZpZ on X.

3. Show that the set XZpZ of elements x P X fixed by every element of ZpZ (that is x P XZpZ if@j P ZpZ, j ¨ x “ x) is the set tpx, ..., xq P Gp, x “ e or x is of order pu. Conclude.

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2 Group presentations

Before getting into the true geometric part of geometric group theory, let’s give a look at the combinatorialapproach. The basic idea is that one can find out everything about a group thanks to a set of generators andtheir relations. Relations are roughly speaking the simplifications of words written in those generators. Forexample, in an abelian group, the word aba´1 can be simplified into the word b. Therefore, one has the relationaba´1 “ b, better written (as we shall see) as aba´1b´1 “ e.

2.1 Free groups

The cornerstone on which is built the presentation theory is the concept of free groups. We have not yetproperly introduced what relations are, but one can understand free groups as groups with no relations (excepttrivial ones such as aa´1 “ e).

Let S be any set. One has to think of S as a generating set of some group G, not yet defined. Define S´1

to be a disjoint copy of S, meaning that there is a bijective map s P S ÞÑ s´1 P S´1. Define some element esuch that e R S, e R S´1. Elements of S´1 should be seen as formal inverses of elements of S in G and e shouldbe seen as the neutral element of G. Also define S :“ S \ S´1 \ teu.

Definition 2.1.1. A word in S is a finite sequence of elements of S. Such a word shall be written w “ sε11 ...sεnn ,

where si P S and εi “ ˘1 (s´1i is actually the image of si by the map s ÞÑ s´1) OR si “ e and εi “ 0.

A formal power of an element of S is an element of S together with a positive integer n. Such a formalpower shall be denoted by sn. A formal power of an element of S´1 is an element of S´1 together with apositive integer n. It shall be denoted by s´n. A formal power of an element of S either is e or is a formalpower of an element of S or of S´1. We shall identify a power ps´1, nq, n ě 1 with ps,´nq. Therefore, one cansee a formal power of an element of S (that is not e) as an element of S together with a non-zero integer.

Definition 2.1.2. A reduced word in S is a finite sequence of powers of elements of S w “ sα11 ...sαn

n wheresi P S, αi P Zzt0u and si ‰ si`1 OR w “ e. Denote the set of reduced words in S by W pSq.

A reduced word should be thought of as an element of a group G written with elements of a generatingset S with all trivial simplifications: eliminating the neutral element every time it appears and eliminating theelements ss´1.

Let’s now give some meaning to that informal explanation. Let G be a group and let S be a generating set.In the following, to simplify notations, we shall always assume that S X S´1 “ H. Define the map

Φ : W pSq Ñ Gw “ sα1

1 ...sαnn ÞÑ sα1

1 ...sαnn

where the element sα11 ...sαn

n in G is just the multiplication of elements of S and inverses of elements of S. Themap Φ is onto, since S is a generating set.

Definition 2.1.3. A free group is a group G together with a generating set S such that the map Φ isone-to-one. We say that S is a free basis of G.

Theorem 2.1.4. Let S be a set. There exists a unique (up to isomorphism) free group of free basis S. It is(abusively) denoted by F pSq. Furthermore, if S and S1 are in bijection, then F pSq is isomorphic to F pS1q. IfS is finite, of cardinal n, then F pSq shall (again, abusively) be denoted by Fn.

One can see S as a subset of W pSq. Since the map Φ is one-to-one, its restriction to S is also one-to-one.This restriction shall be denoted by ΦS : S ãÑ F pSq.

Proof. To prove this theorem, one should take F pSq as the set of reduced words W pSq and define an operationas follows. If w1 and w2 are two reduced words, first define w1w2 to be the concatenation of the words w1 andw2. If the last letter of w1 is the same as the first letter of w2, then simplify w1w2 by eliminating those andkeep on doing that until the word is reduced. Define w1w2 to be this reduced word. To actually show that

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this operation exists and defines a group structure on W pSq is rather long and fastidious. We shall use a trick,often called the van der Waerden trick.

Let G be the group SpW pSqq, that is the group of bijective maps W pSq ÑW pSq. We shall define F pSq asa subgroup of G. Let w “ sα1

1 ...sαnn P W pSq. We say that w starts with s P S if s1 “ s and α1 ą 0. In that

case, we write w “ sw1 where w1 “ sα1´11 ...sαn

n if α1 ą 1 and w1 “ sα22 ...sαn

n otherwise. Similarly, we say thatw starts with s´1 P S´1 if s1 “ s and α1 ă 0. In that case we write w “ s´1w1.

If s P S, define σpsqpwq “ sw if w does not start with s and σpsqpwq “ w1 if w starts with s and w “ sw1.In both cases, σpsqpwq is a reduced word. If s´1 P S´1, define σps´1qpwq “ s´1w if w does not start with s´1

and σps´1qpwq “ w1 if w starts with s´1 and w “ s´1w1. By definition, σpsq ˝ σps´1q “ σps´1q ˝ σpsq “ Id sothat σpsq, σps´1q P G. Moreover, the map σ : S Ñ G is one-to-one and we shall now identify S with σpSq. LetF pSq be the subgroup of G generated by S.

Let’s show that F pSq is free of free basis S. Let w “ sα11 ...sαn

n be a reduced words in S and Φpsα11 ...sαn

n q “ gbe its image in F pSq. Assume that w ‰ e and prove that g ‰ Id P F pSq. In fact, since g P SpW pSqq, we canevaluate g on e, seeing e as an element of W pSq. A simple induction on |α1| ` ... ` |αn| shows that gpeq “ wSince w ‰ e, g ‰ Id.

This concludes the existence part of the theorem. For uniqueness, assume that G and G1 are free of freebasis S. Then there are bijective maps Φ : W pSq Ñ G and Φ1 : W pSq Ñ G1. The map Φ´1 ˝ Φ1 is in fact agroup morphism from G1 to G. Since it is bijective, it is an isomorphism.

Finally, if S and S1 are in bijection, then so are W pSq and W pS1q, so that a reduced word in S is a reducedword in S1. Since all we did was defined up to bijective maps, F pSq and F pS1q are isomorphic.

A powerful tool for proving that some group G is a free group is the ping-pong lemma.

Theorem 2.1.5 (Ping-pong lemma). Let G be a group acting on a set X. Let S be a generating set of G.Assume that there are non-empty disjoint sets pXsqsPS such that

@s, t P S, s ‰ t,@n P Z˚, snXt Ă Xs.

Then, the group G is free with free basis S: G » F pSq.

We leave the proof as an exercise.Exercise 2.1.6. To prove the ping-pong lemma, take g “ sα1

1 ...sαnn P G. Assume that g is reduced. It only has

to be shown that g ‰ e. There are two cases. Either sn “ s1 or sn ‰ s1.

1. Assume that s1 “ sn “ s. Denote by Ys “Ť

t‰sXt. Show that if x0 P Ys, then g ¨ x0 P Xs. Deduce thatg ‰ e.

2. Assume that s1 “ s ‰ t “ sn Show that there exists x P G such that x ‰ s´α11 and x´1 ‰ s´αn

n Byconsidering xgx´1, show that g ‰ e.

2.2 A universal property

The following theorem can be used as an alternative definition for F pSq.

Theorem 2.2.1. Let S be a set and F pSq be a free group of free basis S. Let G be a group and let f : S Ñ Gbe a map. Then, there exists a unique group morphism f : F pSq Ñ G such that f ˝ Φs “ f (recall that ΦS isthe restriction to S of the map Φ : W pSq Ñ F pSq).

This theorem can be summarised with the following diagram.

S

F pSq

G

ΦS

f

f

8

Proof. If f ˝ ΦS “ f , then fpwq “ fpsα11 ...sαn

n q “ fps1qα1 ...fpsnq

αn “ fps1qα1 ...fpsnq

αn , so that f is uniquelydetermined by f . Conversely, since Φ : W pSq Ñ F pSq is a bijective map, this formula defines f . It is welldefined because F pSq is free and it is a group morphism.

Such a property is called a universal property in the category language, and F pSq is called a universalobject, since it satisfies a universal property. Be careful! A universal property is NOT a definition. In general,one has to show that a universal object DOES exist. It is not always an easy thing to do.

The converse of this theorem is true.

Theorem 2.2.2. Let F be a group and ϕS : S Ñ F a map such that for every group G and for every mapf : S Ñ G, there exists a unique group morphism f : F Ñ G such that f ˝ ϕS “ f . Then, F pSq is isomorphicto F .

Proof. First use the universal property with F pSq: there exists a (unique) group morphism f1 : F pSq Ñ F suchthat f1 ˝ ΦS “ ϕS . Then, use it with F : there exists a (unique) group morphism f2 : F Ñ F pSq such thatf2 ˝ ϕS “ ΦS . Thus, the morphism g “ f1 ˝ f2 : F Ñ F satisfies g ˝ ϕS “ ϕS . That means that it is a solutionof the universal property applied to F onto F itself. However, the identity also satisfies this universal property.By uniqueness, one finds that f1 ˝ f2 “ Id. Similarly f2 ˝ f1, so that f1 and f2 are isomorphisms.

Remark 2.2.3. Actually, we proved that the isomorphism F pSq Ñ F is unique (since f1 and f2 are unique).

Let G be a group and S be a generating set. Then, the inclusion is a map S Ñ G. Therefore, one deducesfrom the universal property that there exists a group morphism πS : F pSq Ñ G such that πSpsq “ s. Since Sis a generating set, πS is onto, so that, by the first isomorphism theorem,

F pSqKerpπSq » G.

Conclusion: every group is the quotient of some free group!

2.3 Presentations: generating sets and relations

Definition 2.3.1. Let G be a group and let H be a subgroup of G. The subgroup normally generated byH is the smallest subgroup of G containing H. It is denoted by xxHyy.

Exercise 2.3.2. The subgroup normally generated byH is the set of elements of G of the form g1h1g´11 ...gnhng

´1n ,

with gi P G and hi P H.

Definition 2.3.3. If G is a group and X is any subset of G, then the subgroup normally generated by Xis the subgroup normally generated by xXy. It is denoted by xxXyy.

Let G be a group and let S be a generating set. Then, there exists a group morphism πS : F pSq Ñ G andG » F pSqKerpπSq.

Definition 2.3.4. A relation is an element of KerpπSq. A system of relations is a set R Ă KerpπSq such thatxxRyy “ KerpπSq.

If R is a system of relations, then G is entirely determined by S and R up to isomorphism, since G isisomorphic to F pSqxxRyy. We shall denote this fact by G » xS|Ry. Such a writing is called a presentationof G.

Let’s give examples (actually, exercises)!

1. The group of integers Z is generated by one element. Actually, it is free of free basis this element.Therefore, Z » xx|Hy.

2. More generally, F pSq » xS|Hy.

3. The group Z2 has the presentation xx, y|xyx´1y´1y.

9

4. The group Sn has the presentation xx1, ..., xn|x2i , xixjx

´1i x´1

j p|j ´ i| ě 2q, xixi`1xix´1i`1x

´1i x´1

i`1y.

Sometimes, it is easier to write elements of R as equalities. For example, one can write xy “ yx instead ofxyx´1y´1 in the presentation of Z2.

Conversely, let S be any set and R any subset of F pSq. Define xS|Ry to be F pSqxxRyy.Of course, a same group can have as many presentation as one wants. For instance, it suffices to add

a generator s and to add the relation s “ e, or more generally s “ g, for any g P G. Actually, given twopresentations xS1|R1y, xS2|R2y, asking if the two groups are isomorphic is a really difficult question to answer.Another difficult question is whether some group has a finite presentation (meaning that S and R are bothfinite).

This approach to group theory gave birth to what is called combinatorial group theory. It was initiated byWalther Dyck (who invented free groups in 1882). A good reference for combinatorial group theory is [1]. It isnot what those notes are about, but in some sense, it leads to a geometric theory that we shall now explore.

3 Cayley graphs

We saw that a group can be defined with a set of generators and relations between those. From that, weshall introduce some geometric object, namely the Cayley graph of a group. Basically, the idea is to draw thepresentation xS|Ry by connecting vertices with generators and adding relations to connect vertices. It shall bemade precise in a few moment. First, we shall give some general definitions about graph theory.

3.1 Graphs and trees

We only need some naive graph theory in the following. Nevertheless, to do things properly, we introduce heresome vocabulary.

Definition 3.1.1. A graph is a set V of vertices together with a set E Ă V ˆV of edges such that if pv0, v1q P E,then pv1, v0q P E. In that case, we say that v0 and v1 are connected by an edge and that v1 is a neighbour ofv0.

Remark 3.1.2. We made a choice here of doubling every edge. It is not always the case in literature and onecould define graphs without requiring that pv0, v1q P E if and only if pv1, v0q P E. However, it shall be useful inthe following.

In this definition, it seems that our graphs are non-oriented. It is not true! Defining edges like that, onecan orient them as follows.

Definition 3.1.3. If e “ pv1, v2q P E, define B0peq “ v0 and B1peq “ v1 and define e “ pv1, v0q. Call B0peq theorigin of e and B1peq its end.

Thus, we do deal with oriented graphs, but require that every edge is doubled, for later purpose. It isactually really important to deal with oriented graphs, as we shall see.

For example, the first graph below can be defined as V “ tv0, v1u and E “ tpv0, v1q, pv1, v0qu. The secondone can be defined as V 1 “ tv10, v11, v12u and E1 “ tpv10, v10q, pv10, v11q, pv11, v10q, pv11, v12q, pv12, v11qu. Actually, we didnot draw every edge in the figure, for simplicity. We just put one edge between two vertices (that are connectedby an edge). We shall do the same every time we draw graphs. However, as an abstract graph, every edgeis doubled.

v0 v1

‚ ‚

v10

v11

v12

‚

‚

‚

10

Definition 3.1.4. We say that two vertices v, v1 are connected if there exists a finite sequence of verticespv0 “ v, v1, ..., vn “ v1q such that pvi, vi`1q P E. We say that the graph is connected if every two vertices areconnected.

Definition 3.1.5. A loop is a finite sequence of vertices pv0, ..., vnq such that v0 “ vn, pvi, vi`1q P E andvi ‰ vi`2 (meaning that there is no back-tracking).

Definition 3.1.6. A tree is a connected graph without a loop.


1. The n-ary rooted tree is an infinite tree. Every vertex has n ` 1 neighbours except for one vertex thathas n neighbours and that is called the root. Here is a representation of the binary tree.

......

......

It contains an infinite number of copies of itself, and has very rich symmetry.

2. The n-ary regular tree is also an infinite tree. Every vertex has exactly n neighbours. It is denoted byTn. Here is a representation of T4.

¨¨¨

¨¨¨

¨¨¨¨¨¨

3. A complete graph is a graph pV,Eq such that every two vertices are connected by an edge. It can neverbe a tree. Here is a complete graph with three vertices.

11

We defined the structure of graph. Let’s define morphisms that preserve this structure.

Definition 3.1.7. A graph morphism between two graphs pV,Eq and pV 1, E1q is a map f : V Ñ V 1 suchthat if v0 and v1 are connected by an edge in V , then fpv0q and fpv1q are connected by an edge in V 1.

A graph isomorphism is a graph morphism that is bijective. In that case the inverse of the map is alsoa graph morphism. A graph automorphism is a graph isomorphism from one graph to itself.

Definition 3.1.8. Let pV,Eq be a graph and let G be a group that acts on V . The action of V is called agraph action if whenever v and v1 are connected by an edge, for every g P G, g ¨ v and g ¨ v1 are also connectedby an edge. In other words, the bijective map v ÞÑ g ¨ v is a graph automorphism for every g P G.

3.2 Cayley graphs of a group

When we defined the free group F pSq for some generating set S, we required that S and S´1 were disjoint sets,for simplicity. In the following, we shall deal with generating sets of pre-existing groups. Such an assumption isthen impossible: for example, if s P S is of order 2, then s “ s´1. We slightly change our hypothesis and requirethat S “ S´1, meaning that if s P S Ă G, then s´1 P S. We say that the generating set S is symmetric. Wethen work with S directly instead of S “ S Y S´1.

Definition 3.2.1. Let G be a group, and S a symmetric generating set. The Cayley graph of G, associatedto S, is the graph ΓG,S defined as follows:

1. V “ G (every vertex represents an element of G),

2. pg, g1q P E if there exists s P S such that g1 “ gs (we can go from one vertex to another by multiplyingthe first one on the right by a generator).

Remark 3.2.2. Since S is symmetric, it is true that is pg, g1q P E if and only if pg1, gq P E.Remark 3.2.3. One could define a Cayley graph by requiring that pg, g1q P E if there exists s P S such thatg1 “ sg, that is choosing multiplication on the left. As we shall see, the group G acts on its Cayley graph, andwe do want a left action, so that we chose right multiplication in the definition of the Cayley graph.


1. Let G “ Z and choose the generating set S “ t´1, 1u. Then, two integers are connected by and edge inthe Cayley graph if and only if they differ by 1.

‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚. . . . . .-5 -4 -3 -2 -1 0 1 2 3 4 5

2. Let G “ Z again, but choose the generating set S “ t´3,´2, 2, 3u.

‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚

3. Let G “ Z2 and choose the generating set tp´1, 0qp0,´1q, p1, 0q, p0, 1qu.

‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚. . . . . .

‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚

‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚

‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚

‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚

...

...

12

4. Let G be a finite cylic group ZnZ. For example, take G “ Z5Z and choose the generating set t´1, 1u.The Cayley graph can be represented as a regular n-gon.

‚

‚‚

‚‚

5. Take any group G and choose S to be G. The corresponding Cayley graph is a complete graph with setof vertices G.

6. Let G be a free group F pSq and choose the generating set SYS´1. For example, take G “ F2 and denoteby ta, bu a free basis. Then, take S to be ta´1, b´1, a, bu. As we shall see, the corresponding Cayley graphis a tree. The Cayley graph of F2 is actually the tree T4:

aa´1

b

b´1

As illustrated by the examples of Z with two different generating sets, the choice of S really matters.However, in some sense, from a distant point-of-view, the two Cayley graphs look the same. We shall properlystate what we mean by that later.

Notice that the Cayley graph of G is locally finite if and only if the generating set S is finite. We say thatG is finitely generated if there exists a finite generating set S (we already used this definition at the end ofour discussion about Hopfian groups). It does not mean that every generating set is finite (for example, if Gis infinite, you can always choose G as a generating set). It does not even imply that the subgroups of G arefinitely generated!

Proposition 3.2.4. Let G be a group and S be a symmetric generating set and let ΓG,S be the correspondingCayley graph. Then ΓG,S is a tree if and only if there exists S1 Ă S such that S “ S1 \ S1´1 and G is free offree basis S1.

13

Proof. Assume that G » F pS1q. Every Cayley graph is connected, so that one only has to prove that ΓG,Shas no loop. Let pg, gs1, ..., gs1...snq be a path with si P S. If si ‰ s´1

i`1 (no backtracking), the word s1...sn isreduced. Since G is free, gs1...sn ‰ g, otherwise, one could multiply by g´1 and get s1...sn “ e.

Conversely, assume that ΓG,S is a tree. and define S1 to be a subset of S such that S1 X S1´1 “ H andS1 Y S1´1 “ S, so that S1 also generates G. Thus, one has G » F pS1qKerpπ1Sq. Let w P F pS1q be such thatπ1Spwq “ e P G. Assume that w is reduced and choose such a w of minimal length, that is if w “ s1...sn, withsi P S

1 Y S1´1 “ S, then w minimises n. Since πSpwq “ e, the path pe, es1, ..., es1...snq starts and ends withthe same vertex in ΓG,S . If there were more than one vertex, it would be a loop, since the word is reduced(and therefore, there is no backtracking), hence the path is trivial, and w “ e P F pS1q. That means that π1S isone-to-one, so that G » F pS1q.

Remark 3.2.5. What we actually proved here is that loops in the Cayley graph correspond to relations in thegroup. It gives sense to the phrase "drawing the presentation xS|Ry".

Proposition 3.2.6. Let G be a group, S be a symmetric generating set. Let ΓG,S be the corresponding Cayleygraph. The group G acts on ΓG,S by g ¨v “ gv. It is a free graph action. Moreover for every vertices v, v1 P ΓG,S,there exists g P G such that g ¨ v “ v1 (the action is called transitive).

Proof. The action on vertices is the same as the left translation action of G on itself. Thus, it is free. Ifv, v1 P G, then they are connected by an edge if and only if there exists s P S such that v1 “ vs. In thatcase, g ¨ v1 “ gv1 “ gvs “ pg ¨ vqs, so that g ¨ v and g ¨ v1 are also connected by an edge. If v, v1 P G, thenv1v´1 ¨ v “ v1.

3.3 Metric structures on graphs

To properly state that two Cayley graphs of the same group with different choices of generating sets do lookthe same, we have introduce some geometric vocabulary. First we realise graphs as (geo)metric spaces. Recallthat if e “ pv, v1q is an edge, then e is the edge pv1, vq, B0peq “ v and B1peq “ v1.

Let Γ “ pV,Eq be a graph. The geometric realisation of Γ is the space

G “ pV \ E ˆ r0, 1sqpe, 0q „ B0peq, pe, 1q „ B1peq, pe, tq „ pe, 1´ tq.

Take the discrete topology on V , the discrete topology on E, the product topology on E ˆ r0, 1s and the sumtopology on V \ E ˆ r0, 1s. Finally take the quotient topology on G. This is called the cellular topology ofG.

Define also a distance on G as follows. Declare that each edge is isometric to r0, 1s, meaning that ifx “ pe, tq P G, y “ pe, sq P G, then dpx, yq “ |t ´ s| “ dpy, xq. If γ : r0, 1s Ñ G is continuous for the cellulartopology, then γpr0, 1sq is compact. Therefore, there is only a finite number of vertices v in γpr0, 1sq. Denotethem by v1 “ γpt1q,...,vk “ γptkq. Moreover, γp0q is of the form pe, t0q and v1 of the form pe, 1q or pe, 0q withthe same e, so that dpγp0q, v1q is well defined. Similarly, dpvk, γp1qq is well defined. The length of γpr0, 1sq isthen k ` dpγp0q, v1q ` dpvk, γp1qq. If x, y P G, define the graph distance between x and y to be

dpx, yq “ inftlengthpγq, γ : r0, 1s Ñ G continuous, γp0q “ x, γp1q “ yu.

The topology defined by that distance is weaker than the cellular topology and is not always the same. Itis if the graph is locally finite. Anyway, the topology we shall use is the one induced by this distance.

If G is a group, S a symmetric generating set and if ΓG,S is the corresponding Cayley graph, then G can beembedded into its Cayley graph, identifying an element of G with the corresponding vertex. The restriction toG of the graph distance is called the word metric. The distance between g, g1 P G is then exactly the minimalnumber of generators s1, ..., sn P S that one needs to write g1 “ gs1...sn.

Exercise 3.3.1. Prove that it is indeed a distance.

We now give a definition that measures the "closeness" of two metric spaces. First recall the following.

14

Definition 3.3.2. Let pX, dq, pX 1, d1q be two metric spaces and let f : pX, dq Ñ pX 1, d1q be a map. The mapf is called an isometry if

@x, y P X, dpx, yq “ d1pfpxq, fpyqq.

Such a map is always one-to-one. Moreover, if it is onto, then its inverse is also an isometry.

Definition 3.3.3. Two metric spaces pX, dq and pX 1, d1q are called isometric if there exists an onto isometryf : X Ñ X 1.

Two metric spaces are "close" if there almost exists an onto isometry between them:

Definition 3.3.4. Let pX, dq, pX 1, d1q be two metric spaces and let f : pX, dq Ñ pX 1, d1q be a map. The mapf is called a quasi-isometry if there exists λ ě 1 and c ě 0 such that

@x, y P X,1

λdpx, yq ´ c ď d1pfpxq, fpyq ď λdpx, yq ` c.

Definition 3.3.5. Let pX, dq, pX 1, d1q be two metric spaces and let f : pX, dq Ñ pX 1, d1q be a map. The mapf is called quasi-onto if there exists D ě 0 such that

@x1 P X 1, Dx P X, d1pfpxq, x1q ď D.

Up to some thickening, a quasi-isometry is an isometry and a quasi-onto map is onto.

Lemma 3.3.6. If there exists a quasi-onto quasi-isometry f : X Ñ X 1, then there exists a quasi-onto quasi-isometry g : X 1 Ñ X.

Definition 3.3.7. Twp metric spaces pX, dq and pX 1, d1q are called quasi-isometric if there exists a quasi-ontoquasi-isometry f : X Ñ X 1.

Lemma 3.3.8. This defines an equivalence relation on metric spaces

Exercise 3.3.9. Prove lemma 3.3.6 and lemma 3.3.8.Remark 3.3.10. The action of G on ΓG,S is by isometries, meaning that for every g P G v ÞÑ g ¨ v is an isometryof ΓG,S . Since the action is free, one can see G as a subgroup of the set of isometries of ΓG,S (check that thisis a group!). It is not in general the whole group of isometries. Therefore, in some sense, ΓG,S has much moresymmetry than G.

3.4 Equivalence of Cayley graphs

It is now possible to prove that two Cayley graphs with different (finite) generating sets look the same.Let G be a group and S a generating set. Let ΓG,S be the corresponding Cayley graph. Then the inclusion

of G into ΓG,S is a quasi-onto quasi-isometry. It is actually an isometry since the word metric is the restrictionof the graph metric of ΓG,S . It is quasi-onto because every point on an edge is at distance at most 1 of a vertex.

Theorem 3.4.1. Let G be a finitely generated group and let S, S1 be two finite symmetric generating sets.Denote by d, d1 the corresponding word metrics. Then Id : pG, dq Ñ pG, d1q is a quasi-isometry.

Proof. Let Λ “ maxtdpe, s1q, s1 P S1u and Λ1 “ maxtd1pe, sq, s P Su. If d1pg, hq “ n, then h “ gs11...s1n, s1i P S

1.Write s1i as a word wi in S so that h “ gw1...wn. Then dpg, hq “ dpe, w1...wnq ď Λn “ Λd1pg, hq. Similarly,d1pg, hq ď Λ1dpg, hq. Let λ “ maxpΛ,Λ1q. Then,

1

λdpg, hq ď d1pg, hq ď λdpg, hq.

Corollary 3.4.2. Let G be a finitely generated group and let S, S1 be two finite symmetric generating sets.Let ΓG,S and ΓG,S1 be the corresponding Cayley graphs. Then, for the graph metrics, ΓG,S and ΓG,S1 arequasi-isometric.

15

References

[1] Roger C. Lyndon and Paul E. Schupp. Combinatorial Group Theory. Reprint of the 1977 edition. Classicsin Mathematics. Springer, 2001.

[2] John Meier. Groups, Graphs and Trees. London Mathematical Society Student Texts 73. Cambridge Uni-versity Press, 2008.

[3] Joseph J. Rotman. An Introduction to the Theory of Groups. Graduate Texts in Mathematics. Springer,1995.

[4] Jean-Pierre Serre. Arbres, amalgames, SL2. Third edition. Astérisque 46. Société Mathématique de France,1983.

[5] Jean-Pierre Serre. Trees. Translated by J. Stillwell. Springer, 1980.

16

EXERCISES

Elementary theory of groups

To get warmed up

Exercise 1 Let G be a group.a) Show that the neutral element e is unique.b) Show that if a P G, then its inverse a´1 is unique.

Exercise 2 Show that if G is a group and H is a subgroup of G, then H is itself a group.

Exercise 3 Let G,G1 be two groups and f : GÑ G1 a group morphism.a) Show that fpeq “ e1

b) Show that for every a P G, fpa´1q “ fpaq´1.

Exercise 4 Let G,G1 be two groups and f : GÑ G1 a group morphism. Show that the map f is one-to-oneif and only if Kerpfq “ teu.

Exercise 5 Let G be a group. Denote by ZpGq the set of elements g P G that commute with every elementh of G, i.e. g P ZpGq if @h P G, gh “ hg. Show that ZpGq is a subgroup of G. It is called the center of G.

Exercise 6 Let G be a group and let g, h P G.a) Show that pghq´1 “ h´1g´1.b) Show that pghg´1q´1 “ gh´1g´1.c) Show that for all n P Z, pghg´1qn “ ghng´1.

Normal subgroups and quotient groups

Exercise 7 Show that if G is abelian, then all its subgroups are normal.

Exercise 8 Denote by GLnpKq the set of invertible matrices of size n and by SLnpKq the set of thosematrices with determinant 1.

a) Show that SLnpKq is a normal subgroup of GLnpKq.b) Show that the quotient group GLnpKqSLnpKq is isomorphic to K˚.

Exercise 9 Let G,G1 be two groups and f : G Ñ G1 a group morphism. Show that Kerpfq is a normalsubgroup of G.

Exercise 10 Let G be a group and H a subgroup of G of index 2. Show that H is a normal subgroup of G.

Exercise 11 Let G,G1 be two groups and f : G Ñ G1 a group morphism. Assume that G is finite. Showthat |G| “ |Kerpfq| ˆ |Impfq|.

17

Cyclic groups

Exercise 12a) If n P Z, denote by nZ the set of integers of the form nk. Show that nZ is a normal subgroup of Z.b) Determine all the subgroups of Z.

Hint: Use euclidean division.

Exercise 13 Let G be a cyclic group.a) Show that there exists a group morphism f : ZÑ G that is onto.b) Show that either G is isomorphic to Z or there exists n P Z such that G is isomorphic to ZnZ.

Exercise 14 Find all the automorphisms of Z.

The derived subgroup

Exercise 15 Let G be a group. If g, h P G, define their commutator by rg, hs :“ g´1h´1gh. The derivedsubgroup (sometimes called commutator subgroup) of G is the subgroupDpGq generated by all the commutatorsrg, hs, for g, h P G.

a) Show that the derived subgroup of G is normal.Hint: Use exercise 6.b) Show that GDpGq is abelian.c) Show that DpGq is the smallest normal subgroup of G such that the quotient is abelian.

Group actions

Exercise 16 Let G be a group of order pk, acting on a finite set X. Denote by XG the set of (globally) fixedelements, i.e. x P XG if @g P G, g ¨ x “ x. Show that |X| “ |XG| modulo p.

Hint: Use the class formula.

Exercise 17 [A theorem of Cauchy] Let G be a finite group and let p be a prime number such that p divides|G|. We are going to prove that there exists in G some element g of order exactly p: gp “ e and gk ‰ e, ifk ď p´ 1.

a) Denote by X the set tpg1, ..., gpq P Gp, g1...gp “ eu. Show that |X| “ |G|p´1. Thus, p divides |X|.

b) Show that j ¨ pg1, ..., gpq :“ pg1`j , ..., gp`jq defines a group action of ZpZ on X.c) Show that XZpZ “ tpx, ..., xq P Gp, x “ e or x is of order pu and conclude, using exercise 16.

Exercise 18a) Let G be a group of order p. Show that G is isomorphic to ZpZ.b) Let G be a group of order pk. Using exercise 16, show that ZpGq is not trivial.c) Let G be a group of order p2. Show that G is abelian.

Exercise 19 [A Burnside formula] Let G be a finite group acting on a finite set X. Denote by Ω the set ofall orbits. Denote by A the set tpg, xq P GˆX, g ¨ x “ xu.

a) Show that |A| “ř

ωPΩ

ř

xPω |Stabx| “ |Ω||G|.b) If g P G, denote by Fixg the set of elements x P X such that g ¨ x “ x. Show that |A| “

ř

gPG |Fixg|.c) Conclude that |Ω| “ 1|G|

ř

gPG |Fixg|.

18

EXERCISES

Group presentations

Free groups

Exercise 1 [The Ping-pong lemma] We are going to prove a very helpful lemma. It is a powerful tool to provethat some groups are free, another efficient tool being the Bass-Serre theory, that is actions of groups on trees.

Lemma. Let G be a group acting on a set X. Let S be a generating set of G. Assume that there are non-emptydisjoint sets pXsqsPS such that

@s, t P S, s ‰ t,@n P Z˚, snXt Ă Xs.

Then, the group G is free with free basis S: G » F pSq.

To prove this lemma, take g “ sα11 ...sαn

n P G. Assume that g is reduced. It only has to be shown that g ‰ e.There are two cases. Either sn “ s1 or sn ‰ s1.

a) Show that each si is of infinite order, i.e. sni “ e if and only if n “ 0.b) Assume that s1 “ sn “ s. Denote by Ys “

Ť

t‰sXt. Show that if x0 P Ys, then g ¨ x0 P Xs. Deducethat g ‰ e.

c) Assume that s1 “ s ‰ t “ sn Show that there exists x P G such that x ‰ s´α11 and x´1 ‰ s´αn

n Byconsidering xgx´1, show that g ‰ e.

Exercise 2 Let F pa, bq a free group of rank 2 and let Sn be the set tb, aba´1, ..., an´1ban´1u. For k ě 0,denote by Xk the set of reduced words of F2 beginning with akb or akb´1.

a) Show that the subgroup of F2 generated by Sn acts faithfully on F2.b) Using the ping-pong lemma, show that the subgroup of F2 generated by Sn is free of rank n.c) Show that F2 is a subgroup of Fn, for every n ě 2.

This shows that Fk is a subgroup of Fn for every k ě 1 and n ě 2. However, the following exercise showsthat Fk fi Fn, if k ‰ n.

Exercise 3 Recall the definition from exercise 15 in the last exercise sheet: the derived subgroup of a groupG is the smallest normal subgroup of G such that the quotient is abelian. It is generated by elements of theform rg, hs :“ g´1h´1gh (those elements are called commutators). The derived subgroup of G is denoted byDpGq.

a) Show that F pSqDpF pSqq » ZS (see exercise 5 below). You now understand why the groups Zk arecalled abelian free groups.

b) Show that if F pSq » F pS1q then there exists a bijection between S and S1. In particular, if Fn » Fk,then n “ k.

Relations

Exercise 4 Let G be a group and H be a subgroup of G. Recall that xxHyy is the smallest normal subgroupof G containing H. Show that xxHyy is the set of elements of G of the form g1h1g

´11 ...gnhng

´1n .

Exercise 5 Prove that Zn » xs1, ..., sn|sisj “ sjsiy.

19

Exercise 6 Denote by Sn the group of permutations of t1, ..., nu. Show that

Sn » xx1, ..., xn|x2i , xixj “ xjxi, pj ´ i ě 2q, xixi`1xi “ xi`1xixi`1y.

A geometric realisation of F2.

Exercise 7 This exercise is for those who know a little about hyperbolic geometry. However, someonewithout knowledge in this domain of mathematics can still do the exercise, but will miss the geometric flavour.It seems that it is the first example of the use of the ping-pong lemma to show that some group is free.

Take the hyperbolic upper-half plane H2, i.e. H2 :“ tz “ x ` iy P C, x P R, y ą 0u. Recall that the groupSL2pRq acts on H2 by homography, that is

ˆ

a bc d

˙

¨ z “az ` b

cz ` d.

This action is continuous on H2 :“ tz “ x` iy, x P R, y ě 0u Y t8u (the closure is taken into the Riemannsphere). By definition, one has

ˆ

a bc d

˙

¨ 8 “a

c.

a) Define BH2 :“ tz “ x P Ru Y t8u. Show that the action preserves BH2, i.e. if z P BH2, then its imageby any matrix of SL2pRq is still in BH2.

b) Consider the two matrices

A “

ˆ

2 00 12

˙

and B “ˆ?

2 1

1?

2

˙

.

Show that A,B P SL2pZq and that A and B only have two fixed points on BH2. Denote them by a1, a2, b1, b2,with A ¨ ai “ ai, B ¨ bi “ bi.

c) Show that for any z P H2, An ¨ z converges to a1 if n tends to `8 and converges to a2 when n tendsto ´8.

d) Show that for any z P H2, Bn ¨ z converges to b1 if n tends to `8 and converges to b2 when n tendsto ´8.

e) Using the ping-pong lemma, show that for some integers k, l ě 1, the subgroup of SL2pZq generatedby Ak and Bl is free.

This result is really important in hyperbolic geometry. The matrices A and B are called loxodromic (some-times they are called hyperbolic). Since their fixed points are disjoint, they are called independent loxodromicmatrices. Such free groups generated by independent loxodromic matrices are called Schottky groups.

20

EXERCISES

Cayley graphs

Words and metric structures

Exercise 1 Recall that if G is a group and S a generating set, we define dpg, hq to be the minimum numberof generators s1, ..., sn of S such that g´1h “ s1...sn. Show that d defines a metric structure on G (it is calledthe word metric).

Exercise 2 Recall that a quasi-isometry from a metric space pX, dq to a metric space pX 1, d1q is a mapf : X Ñ X 1 with the property that there exists λ ě 1 and c ě 0 such that for every x, y P X, 1

λdpx, yq ´ c ďd1pfpxq, fpyqq ď λdpx, yq ` c.

Recall that a map f : pX, dq Ñ pX 1, d1q is quasi-onto if there exists α ě 0 such that for every x1 P X 1 thereexists x P X such that dpfpxq, x1q ď α.

a) Show that if there exists a quasi-onto quasi-isometry f : X Ñ X 1, then there exists quasi-onto quasi-isometry g : X 1 Ñ X.

b) Say that two metric spaces pX, dq, pX 1, d1q are quasi-isometric if there exists a quasi-onto quasi-isometryf : X Ñ X 1. Show that this is an equivalence relation. Show that you cannot remove the quasi-onto hypothesis.

Drawing Cayley graphs

Exercise 3 Define the dihedral groupDn to be the group of symmetries of the regular n-gon, that is reflectionsand rotations of the complex-plane that preserves the regular n-gon. Fix the vertices of the regular n-gon tobe the set tz P C, |z|n “ 1u. Define σ to be the reflection of axis the real line, that is σpzq “ z. Define ρn to bethe rotation of angle 2πn.

a) Show that σρnσ´1 “ ρ´1n .

b) Show that Dn is generated by σ and ρn and that Dn is of order 2n.c) Choose your favorite n and draw the Cayley graph of Dn with respect to the generating set tρn, σu.

Exercise 4 Draw the Cayley graph of Z5Z‘ Z2Z with respect to the generating set tp1, 0q, p0, 1qu. Whatis the difference between this graph and the Cayley graph of D5 with respect to tρ5, σu ?

Exercise 5 Recall that there is an embedding of F3 into F2: if F2 is generated by ta, bu, then the subgroupof F2 generated by tb, aba´1, a2ba´2u is isomorphic to F3. Try to draw this embedding, that is try to find theCayley graph of F3 inside the Cayley graph of F2.

Exercise 6 If Γ is a graph and e P Γ an edge and if G is a group acting on Γ, there are two natural ways ofdefining the stabiliser of e. Define Stabe “ tg P G, g ¨ e “ eu and Stab1e “ tg P G, g ¨ v1 “ v1, g ¨ v2 “ v2u, withe “ pv1, v2q. Let Γ be the "wheel graph" drawn below. Let G be the symmetry group of Γ, that is the groupof isometries of the plan that preserves Γ. Show that if e is a central edge, then Stabe » Stab1e » Z2Z, but ife is an external edge, then Stabe » Z2Z, while Stab1e “ tIdu.

‚

‚

‚‚

‚

‚

‚

The Wheel graph

21

An introduction to geometric group theory Pristina, March 2017

EXERCISES


To get warmed up














Exercise 11 Let G,G1 be two groups and f : GÑ G1 a group morphism. Assume that G is finite. Show that|G| “ |Kerpfq| ˆ |Impfq|.

1


Cyclic groups


Hint : Use euclidean division.





a) Show that the derived subgroup of G is normal.Hint : Use exercise 6.

b) Show that GDpGq is abelian.c) Show that DpGq is the smallest normal subgroup of G such that the quotient is abelian.

Group actions


Hint : Use the class formula.

Exercise 17 [A theorem of Cauchy] Let G be a finite group and let p be a prime number such that p divides|G|. We are going to prove that there exists in G some element g of order exactly p : gp “ e and gk ‰ e, ifk ď p´ 1.






ωPΩ

ř


ř


ř

gPG |Fixg|.

2


EXERCISES


To get warmed up















1


Cyclic groups









Group actions









ωPΩ

ř


ř


ř

gPG |Fixg|.

2


EXERCISES


To get warmed up















1


Cyclic groups









Group actions









ωPΩ

ř


ř


ř

gPG |Fixg|.

2

an introduction to geometric group theory pristina -...

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