Combinatorial group theory afFrom Wikipedia, the free encyclopedia

Contents

1 Absolute presentation of a group 11.1 Formal Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 AndrewsCurtis conjecture 32.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Automorphism group of a free group 43.1 Presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

4 BaumslagSolitar group 64.1 Linear representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.2 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

5 Combinatorial group theory 95.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

6 Commutator collecting process 106.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

7 Cyclically reduced word 117.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

8 Fox derivative 128.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

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ii CONTENTS

9 Free group 149.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159.3 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159.4 Universal property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169.5 Facts and theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179.6 Free abelian group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179.7 Tarskis problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

10 Freiheitssatz 2010.1 Statement of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2010.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2010.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2010.4 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 21

10.4.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2110.4.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2110.4.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Chapter 1

Absolute presentation of a group

In mathematics, one method of dening a group is by an absolute presentation.[1]

Recall that to dene a groupG by means of a presentation, one species a set S of generators so that every elementof the group can be written as a product of some of these generators, and a setR of relations among those generators.In symbols:

G ' hS j Ri:

InformallyG is the group generated by the set S such that r = 1 for all r 2 R . But here there is a tacit assumptionthat G is the freest such group as clearly the relations are satised in any homomorphic image of G . One way ofbeing able to eliminate this tacit assumption is by specifying that certain words in S should not be equal to 1: Thatis we specify a set I , called the set of irrelations, such that i 6= 1 for all i 2 I .

1.1 Formal DenitionTo dene an absolute presentation of a groupG one species a set S of generators, a setR of relations among thosegenerators and a set I of irrelations among those generators. We then say G has absolute presentation

hS j R; Ii:

provided that:

1. G has presentation hS j Ri:

2. Given any homomorphism h : G ! H such that the irrelations I are satised in h(G) , G is isomorphic toh(G) .

A more algebraic, but equivalent, way of stating condition 2 is:

2a. if N / G is a non-trivial normal subgroup of G then I \N 6= f1g :

Remark: The concept of an absolute presentation has been fruitful in elds such as algebraically closed groups and theGrigorchuk topology. In the literature, in a context where absolute presentations are being discussed, a presentation(in the usual sense of the word) is sometimes referred to as a relative presentation. The term seems rather strangeas one may well ask relative to what?" and the only justication seems to be that relative is habitually used as anantonym to absolute.

1

2 CHAPTER 1. ABSOLUTE PRESENTATION OF A GROUP

1.2 ExampleThe cyclic group of order 8 has the presentation

ha j a8 = 1i:

But, up to isomorphism there are three more groups that satisfy the relation a8 = 1 namely:

ha j a4 = 1i

ha j a2 = 1iha j a = 1i:However none of these satisfy the irrelation a4 6= 1 . So an absolute presentation for the cyclic group of order 8 is:

ha j a8 = 1; a4 6= 1i:

It is part of the denition of an absolute presentation that the irrelations are not satised in any proper homomorphicimage of the group. Therefore:

ha j a8 = 1; a2 6= 1i

Is not an absolute presentation for the cyclic group of order 8 because the irrelation a2 6= 1 is satised in the cyclicgroup of order 4.

1.3 BackgroundThe notion of an absolute presentation arises from Bernhard Neumann's study of the isomorphism problem foralgebraically closed groups.[1]

Acommon strategy for considering whether two groupsG andH are isomorphic is to consider whether a presentationfor one might be transformed into a presentation for the other. However algebraically closed groups are neither nitelygenerated nor recursively presented and so it is impossible to compare their presentations. Neumann considered thefollowing alternative strategy:Suppose we know that a group G with nite presentation G = hx1; x2 j Ri can be embedded in the algebraicallyclosed group G then given another algebraically closed groupH , we can ask Can G be embedded inH ?"It soon becomes apparent that a presentation for a group does not contain enough information to make this decisionfor while there may be a homomorphism h : G ! H , this homomorphism need not be an embedding. What isneeded is a specication for G that forces any homomorphism preserving that specication to be an embedding.An absolute presentation does precisely this.

1.4 References[1] B. Neumann, The isomorphism problem for algebraically closed groups, in: Word Problems, Decision Problems, and the

Burnside Problem in Group Theory, Amsterdam-London (1973), pp. 553562.

Chapter 2

AndrewsCurtis conjecture

In mathematics, the AndrewsCurtis conjecture states that every balanced presentation of the trivial group canbe transformed into a trivial presentation by a sequence of Nielsen transformations on the relators together withconjugations of relators, named after James J. Andrews and Morton L. Curtis who proposed it in 1965. It is dicultto verify whether the conjecture holds for a given balanced presentation or not.It is widely believed that the AndrewsCurtis conjecture is false. While there are no counterexamples known, thereare numerous potential counterexamples.[1] It is known that the Zeeman conjecture on collapsibility implies theAndrewsCurtis conjecture.[2]

2.1 References Andrews, J. J.; Curtis, M. L. (1965), Free groups and handlebodies, Proceedings of the American Mathemat-

ical Society (American Mathematical Society) 16 (2): 192195, doi:10.2307/2033843, JSTOR 2033843, MR0173241

Hazewinkel, Michiel, ed. (2001), Low-dimensional topology, problems in, Encyclopedia of Mathematics,Springer, ISBN 978-1-55608-010-4

[1] Open problems in combinatorial group theory

[2] Hazewinkel, Michiel, ed. (2001), Collapsibility, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

3

Chapter 3

Automorphism group of a free group

In mathematical group theory, the automorphism group of a free group is a discrete group of automorphisms ofa free group. The quotient by the inner automorphisms is the outer automorphism group of a free group, which issimilar in some ways to the mapping class group of a surface.

3.1 PresentationNielsen (1924) showed that the automorphism dened by the elementary Nielsen transformations generates the fullautomorphism group of a nitely generated free group. Nielsen, and later Bernhard Neumann used these ideas togive nite presentations of the automorphism groups of free groups. This is also described in (Magnus, Karrass &Solitar 2004, p. 131, Th 3.2).The automorphism group of the free group with ordered basis [ x1, , xn ] is generated by the following 4 elementaryNielsen transformations:

Switch x1 and x2 Cyclically permute x1, x2, , xn, to x2, , xn, x1. Replace x1 with x11

Replace x1 with x1x2

These transformations are the analogues of the elementary row operations. Transformations of the rst two kinds areanalogous to row swaps, and cyclic row permutations. Transformations of the third kind correspond to scaling a rowby an invertible scalar. Transformations of the fourth kind correspond to row additions.Transformations of the rst two types suce to permute the generators in any order, so the third type may be appliedto any of the generators, and the fourth type to any pair of generators.Nielsen gave a rather complicated nite presentation using these generators, described in (Magnus, Karrass & Solitar2004, p. 165, Section 3.5).

3.2 References Magnus, Wilhelm; Abraham Karrass, Donald Solitar (2004), Combinatorial Group Theory, New York: DoverPublications, ISBN 978-0-486-43830-6, MR 0207802

Nielsen, Jakob (1921), Om regning med ikke-kommutative faktorer og dens anvendelse i gruppeteorien,Math. Tidsskrift B, (in Danish) 1921: 7894, JFM 48.0123.03

Nielsen, Jakob (1924), Die Isomorphismengruppe der freien Gruppen.,Mathematische Annalen (in German)91: 169209, doi:10.1007/BF01556078, JFM 50.0078.04

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3.2. REFERENCES 5

Vogtmann, Karen (2002), Proceedings of the Conference on Geometric and Combinatorial Group Theory,Part I (Haifa, 2000)", Geometriae Dedicata 94: 131, doi:10.1023/A:1020973910646, ISSN 0046-5755, MR1950871 |chapter= ignored (help)

Chapter 4

BaumslagSolitar group

One sheet of the Cayley graph of the BaumslagSolitar group BS(1, 2). Red edges correspond to a and blue edges correspond to b.

In the mathematical eld of group theory, the BaumslagSolitar groups are examples of two-generator one-relatorgroups that play an important role in combinatorial group theory and geometric group theory as (counter)examplesand test-cases. They are given by the group presentation

a; b : bamb1 = an

:

For each integer m and n, the BaumslagSolitar group is denoted BS(m, n). The relation in the presentation is calledthe BaumslagSolitar relation.Some of the various BS(m, n) are well-known groups. BS(1, 1) is the free abelian group on two generators, and BS(1,1) is the fundamental group of the Klein bottle.The groups were dened by Gilbert Baumslag and Donald Solitar in 1962 to provide examples of non-Hopan groups.The groups contain residually nite groups, Hopan groups that are not residually nite, and non-Hopan groups.

4.1 Linear representationDene

6

4.1. LINEAR REPRESENTATION 7

The sheets of the Cayley graph of the Baumslag-Solitar group BS(1, 2) t together into an innite binary tree.

A =

1 10 1

; B =

nm 00 1

:

The matrix group G generated by A and B is a homomorphic image of BS(m, n), via the homomorphism induced by

a 7! A; b 7! B:

8 CHAPTER 4. BAUMSLAGSOLITAR GROUP

It is worth noting that this will not, in general, be an isomorphism. For instance if BS(m, n) is not residually nite(i.e. if it is not the case that |m| = 1, |n| = 1, or |m| = |n|[1]) it cannot be isomorphic to a nitely generated linear group,which is known to be residually nite by a theorem of Mal'cev.[2]

4.2 Notes[1] See Nonresidually Finite One-Relator Groups by Stephen Meskin for a proof of the residual niteness condition

[2] Anatoli Ivanovich Mal'cev, On the faithful representation of innite groups by matrices Transl. Amer. Math. Soc. (2),45 (1965), pp. 118

4.3 References D.J. Collins (2001), BaumslagSolitar group, inHazewinkel, Michiel, Encyclopedia ofMathematics, Springer,ISBN 978-1-55608-010-4

Gilbert Baumslag and Donald Solitar, Some two-generator one-relator non-Hopan groups, Bulletin of theAmerican Mathematical Society 68 (1962), 199201. MR 0142635

Chapter 5

Combinatorial group theory

Inmathematics, combinatorial group theory is the theory of free groups, and the concept of a presentation of a groupby generators and relations. It is much used in geometric topology, the fundamental group of a simplicial complexhaving in a natural and geometric way such a presentation. A very closely related topic is geometric group theory,which today largely subsumes combinatorial group theory, using techniques from outside combinatorics besides.It also comprises a number of algorithmically insoluble problems, most notably the word problem for groups; and theclassical Burnside problem.

5.1 HistorySee (Chandler & Magnus 1982) for a detailed history of combinatorial group theory.A proto-form is found in the 1856 icosian calculus of William Rowan Hamilton, where he studied the icosahedralsymmetry group via the edge graph of the dodecahedron.The foundations of combinatorial group theory were laid by Walther von Dyck, student of Felix Klein, in the early1880s, who gave the rst systematic study of groups by generators and relations.[1]

5.2 References[1] Stillwell, John (2002), Mathematics and its history, Springer, p. 374, ISBN 978-0-387-95336-6

Chandler, B.; Magnus, Wilhelm (December 1, 1982), The History of Combinatorial Group Theory: A CaseStudy in the History of Ideas, Studies in the History of Mathematics and Physical Sciences (1st ed.), Springer,p. 234, ISBN 978-0-387-90749-9

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Chapter 6

Commutator collecting process

In mathematical group theory, the commutator collecting process is a method for writing an element of a group asa product of generators and their higher commutators arranged in a certain order. The commutator collecting processwas introduced by Philip Hall (1934). He called it a collecting process though it is also often called a collectionprocess.

6.1 StatementThe commutator collecting process is usually stated for free groups, as a similar theorem then holds for any group bywriting it as a quotient of a free group.Suppose F1 is a free group on generators a1, ..., am. Dene the descending central series by putting

Fn = [Fn, F1]

The basic commutators are elements of F1 dened and ordered as follows.

The basic commutators of weight 1 are the generators a1, ..., am. The basic commutators of weight w > 1 are the elements [x, y] where x and y are basic commutators whoseweights sum to w, such that x > y and if x = [u, v] for basic commutators u and v then y v.

Commutators are ordered so that x > y if x has weight greater than that of y, and for commutators of any xed weightsome total ordering is chosen.Then Fn/Fn is a fnitely-generated free abelian group with a basis consisting of basic commutators of weight n.Then any element of F can be written as

g = cn11 cn22 cnkk c

where the ci are the basic commutators of weight at most m arranged in order, and c is a product of commutators ofweight greater than m, and the ni are integers.

6.2 References Hall, Marshall (1959), The theory of groups, Macmillan, MR 0103215 Hall, Philip (1934), A contribution to the theory of groups of prime-power order, Proceedings of the London

Mathematical Society 36: 2995, doi:10.1112/plms/s2-36.1.29 Huppert, B. (1967), Endliche Gruppen (in German), Berlin, New York: Springer-Verlag, pp. 9093, ISBN978-3-540-03825-2, MR 0224703, OCLC 527050

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Chapter 7

Cyclically reduced word

In mathematics, cyclically reduced word is a concept of combinatorial group theory.Let F(X) be a free group. Then a wordw in F(X) is said to be cyclically reduced if and only if every cyclic permutationof the word is reduced.

7.1 Properties Every cyclic shift and the inverse of a cyclically reduced word are cyclically reduced again. Every word is conjugate to a cyclically reduced word. The cyclically reduced words are minimal-length repre-sentatives of the conjugacy classes in the free group. This representative is not uniquely determined, but it isunique up to cyclic shifts (since every cyclic shift is a conjugate element).

7.2 References Solitar, Donald; Magnus, Wilhelm; Karrass, Abraham (1976), Combinatorial group theory: presentations of

groups in terms of generators and relations, New York: Dover, pp. 33,188,212, ISBN 0-486-63281-4

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Chapter 8

Fox derivative

In mathematics, the Fox derivative is an algebraic construction in the theory of free groups which bears manysimilarities to the conventional derivative of calculus. The Fox derivative and related concepts are often referred toas the Fox calculus, or (Foxs original term) the free dierential calculus. The Fox derivative was developed in aseries of ve papers by mathematician Ralph Fox, published in Annals of Mathematics beginning in 1953.

8.1 DenitionIf G is a free group with identity element e and generators gi, then the Fox derivative with respect to gi is a functionfrom G into the integral group ring ZG which is denoted @@gi , and obeys the following axioms:

@@gi (gj) = ij , where ij is the Kronecker delta

@@gi (e) = 0

@@gi (uv) = @@gi (u) + u @@gi (v) for any elements u and v of G.

The rst two axioms are identical to similar properties of the partial derivative of calculus, and the third is a modiedversion of the product rule. As a consequence of the axioms, we have the following formula for inverses

@@gi (u1) = u1 @@gi (u) for any element u of G.

8.2 ApplicationsThe Fox derivative has applications in group cohomology, knot theory and covering space theory, among other areasof mathematics.

8.3 See also

Alexander polynomial

Free group

Ring (mathematics)

Integral domain

12

8.4. REFERENCES 13

8.4 References Brown, Kenneth S. (1972). Cohomology of Groups. Graduate Texts in Mathematics 87. Springer Verlag.ISBN 0-387-90688-6. MR 0672956.

Fox, Ralph (May 1953). Free Dierential Calculus, I: Derivation in the Free Group Ring. Annals of Math-ematics 57 (3): 547560. doi:10.2307/1969736. JSTOR 1969736. MR 0053938.

Fox, Ralph (March 1954). Free Dierential Calculus, II: The Isomorphism Problem of Groups. Annals ofMathematics 59 (2): 196210. doi:10.2307/1969686. JSTOR 1969686. MR 0062125.

Fox, Ralph (November 1956). Free Dierential Calculus, III: Subgroups. Annals of Mathematics 64 (2):407419. doi:10.2307/1969592. JSTOR 1969592. MR 0095876.

Chen, Kuo-Tsai; Ralph Fox; Roger Lyndon (July 1958). Free Dierential Calculus, IV: The Quotient Groupsof the Lower Central Series. Annals of Mathematics 68 (1): 8195. doi:10.2307/1970044. JSTOR 1970044.MR 0102539.

Fox, Ralph (May 1960). Free Dierential Calculus, V: The Alexander Matrices Re-Examined. Annals ofMathematics 71 (3): 408422. doi:10.2307/1969936. JSTOR 1969936. MR 0111781.

Chapter 9

Free group

Diagram showing what the Cayley graph for the free group on two generators would look like. Each vertex represents an elementof the free group, and each edge represents multiplication by a or b.

In mathematics, the free group FS over a given set S consists of all expressions (a.k.a. words, or terms) that can bebuilt from members of S, considering two expressions dierent unless their equality follows from the group axioms

14

9.1. HISTORY 15

(e.g. st = suu1t, but s t for s,t,uS). The members of S are called generators of FS. An arbitrary group G is calledfree if it is isomorphic to FS for some subset S of G, that is, if there is a subset S of G such that every element of Gcan be written in one and only one way as a product of nitely many elements of S and their inverses (disregardingtrivial variations such as st = suu1t).A related but dierent notion is a free abelian group, both notions are particular instances of a free object fromuniversal algebra.

9.1 History

Free groups rst arose in the study of hyperbolic geometry, as examples of Fuchsian groups (discrete groups actingby isometries on the hyperbolic plane). In an 1882 paper, Walther von Dyck pointed out that these groups have thesimplest possible presentations.[1] The algebraic study of free groups was initiated by Jakob Nielsen in 1924, whogave them their name and established many of their basic properties.[2][3][4] Max Dehn realized the connection withtopology, and obtained the rst proof of the full NielsenSchreier theorem.[5] Otto Schreier published an algebraicproof of this result in 1927,[6] and Kurt Reidemeister included a comprehensive treatment of free groups in his 1932book on combinatorial topology.[7] Later on in the 1930s, Wilhelm Magnus discovered the connection between thelower central series of free groups and free Lie algebras.

9.2 Examples

The group (Z,+) of integers is free; we can take S = {1}. A free group on a two-element set S occurs in the proof ofthe BanachTarski paradox and is described there.On the other hand, any nontrivial nite group cannot be free, since the elements of a free generating set of a freegroup have innite order.In algebraic topology, the fundamental group of a bouquet of k circles (a set of k loops having only one point incommon) is the free group on a set of k elements.

9.3 Construction

The free group FS with free generating set S can be constructed as follows. S is a set of symbols and we supposefor every s in S there is a corresponding inverse symbol, s1, in a set S1. Let T = S S1, and dene a word in S tobe any written product of elements of T. That is, a word in S is an element of the monoid generated by T. The emptyword is the word with no symbols at all. For example, if S = {a, b, c}, then T = {a, a1, b, b1, c, c1}, and

ab3c1ca1c

is a word in S. If an element of S lies immediately next to its inverse, the word may be simplied by omitting the s,s1 pair:

ab3c1ca1c ! ab3 a1c:

Aword that cannot be simplied further is called reduced. The free group FS is dened to be the group of all reducedwords in S. The group operation in FS is concatenation of words (followed by reduction if necessary). The identityis the empty word. A word is called cyclically reduced, if its rst and last letter are not inverse to each other. Everyword is conjugate to a cyclically reduced word, and a cyclically reduced conjugate of a cyclically reduced word is acyclic permutation of the letters in the word. For instance b1abcb is not cyclically reduced, but is conjugate to abc,which is cyclically reduced. The only cyclically reduced conjugates of abc are abc, bca, and cab.

16 CHAPTER 9. FREE GROUP

9.4 Universal property

The free group FS is the universal group generated by the set S. This can be formalized by the following universalproperty: given any function from S to a group G, there exists a unique homomorphism : FS G making thefollowing diagram commute (where the unnamed mapping denotes the inclusion from S into FS):

That is, homomorphisms FS G are in one-to-one correspondence with functions S G. For a non-free group, thepresence of relations would restrict the possible images of the generators under a homomorphism.To see how this relates to the constructive denition, think of the mapping from S to FS as sending each symbol toa word consisting of that symbol. To construct for given , rst note that sends the empty word to identity of Gand it has to agree with on the elements of S. For the remaining words (consisting of more than one symbol) canbe uniquely extended since it is a homomorphism, i.e., (ab) = (a) (b).The above property characterizes free groups up to isomorphism, and is sometimes used as an alternative denition.It is known as the universal property of free groups, and the generating set S is called a basis for FS. The basis for afree group is not uniquely determined.Being characterized by a universal property is the standard feature of free objects in universal algebra. In the languageof category theory, the construction of the free group (similar to most constructions of free objects) is a functor fromthe category of sets to the category of groups. This functor is left adjoint to the forgetful functor from groups to sets.

9.5. FACTS AND THEOREMS 17

9.5 Facts and theoremsSome properties of free groups follow readily from the denition:

1. Any group G is the homomorphic image of some free group F(S). Let S be a set of generators of G. The naturalmap f: F(S) G is an epimorphism, which proves the claim. Equivalently, G is isomorphic to a quotientgroup of some free group F(S). The kernel of f is a set of relations in the presentation of G. If S can be chosento be nite here, then G is called nitely generated.

2. If S has more than one element, then F(S) is not abelian, and in fact the center of F(S) is trivial (that is, consistsonly of the identity element).

3. Two free groups F(S) and F(T) are isomorphic if and only if S and T have the same cardinality. This cardinalityis called the rank of the free group F. Thus for every cardinal number k, there is, up to isomorphism, exactlyone free group of rank k.

4. A free group of nite rank n > 1 has an exponential growth rate of order 2n 1.

A few other related results are:

1. The NielsenSchreier theorem: Every subgroup of a free group is free.2. A free group of rank k clearly has subgroups of every rank less than k. Less obviously, a (nonabelian!) free

group of rank at least 2 has subgroups of all countable ranks.3. The commutator subgroup of a free group of rank k > 1 has innite rank; for example for F(a,b), it is freely

generated by the commutators [am, bn] for non-zero m and n.4. The free group in two elements is SQ universal; the above follows as any SQ universal group has subgroups of

all countable ranks.5. Any group that acts on a tree, freely and preserving the orientation, is a free group of countable rank (given by

1 plus the Euler characteristic of the quotient graph).6. The Cayley graph of a free group of nite rank, with respect to a free generating set, is a tree on which the

group acts freely, preserving the orientation.7. The groupoid approach to these results, given in the work by P.J. Higgins below, is kind of extracted from an

approach using covering spaces. It allows more powerful results, for example on Grushkos theorem, and anormal form for the fundamental groupoid of a graph of groups. In this approach there is considerable use offree groupoids on a directed graph.

8. Grushkos theorem has the consequence that if a subset B of a free group F on n elements generates F and hasn elements, then B generates F freely.

9.6 Free abelian groupFurther information: free abelian group

The free abelian group on a set S is dened via its universal property in the analogous way, with obvious modications:Consider a pair (F, ), where F is an abelian group and : S F is a function. F is said to be the free abelian groupon S with respect to if for any abelian group G and any function : S G, there exists a unique homomorphismf: F G such that

f((s)) = (s), for all s in S.

The free abelian group on S can be explicitly identied as the free group F(S) modulo the subgroup generated by itscommutators, [F(S), F(S)], i.e. its abelianisation. In other words, the free abelian group on S is the set of words thatare distinguished only up to the order of letters. The rank of a free group can therefore also be dened as the rank ofits abelianisation as a free abelian group.

18 CHAPTER 9. FREE GROUP

9.7 Tarskis problemsAround 1945, Alfred Tarski asked whether the free groups on two or more generators have the same rst order theory,and whether this theory is decidable. Sela (2006) answered the rst question by showing that any two nonabelian freegroups have the same rst order theory, and Kharlampovich & Myasnikov (2006) answered both questions, showingthat this theory is decidable.A similar unsolved (as of 2011) question in free probability theory asks whether the von Neumann group algebras ofany two non-abelian nitely generated free groups are isomorphic.

9.8 See also Generating set of a group Presentation of a group Nielsen transformation, a factorization of elements of the automorphism group of a free group Normal form for free groups and free product of groups Free product

9.9 Notes[1] von Dyck, Walther (1882). Gruppentheoretische Studien (Group-theoretical Studies)". Mathematische Annalen 20 (1):

144. doi:10.1007/BF01443322.

[2] Nielsen, Jakob (1917). Die Isomorphismen der allgemeinen unendlichen Gruppe mit zwei Erzeugenden. MathematischeAnnalen 78 (1): 385397. doi:10.1007/BF01457113. JFM 46.0175.01. MR 1511907.

[3] Nielsen, Jakob (1921). On calculation with noncommutative factors and its application to group theory. (Translated fromDanish)". The Mathematical Scientist. 6 (1981) (2): 7385.

[4] Nielsen, Jakob (1924). Die Isomorphismengruppe der freien Gruppen. Mathematische Annalen 91 (3): 169209.doi:10.1007/BF01556078.

[5] See Magnus, Wilhelm; Moufang, Ruth (1954). Max Dehn zum Gedchtnis. Mathematische Annalen 127 (1): 215227.doi:10.1007/BF01361121.

[6] Schreier, Otto (1928). Die Untergruppen der freien Gruppen. Abhandlungen aus dem Mathematischen Seminar derUniversitt Hamburg 5: 161183. doi:10.1007/BF02952517.

[7] Reidemeister, Kurt (1972 (1932 original)). Einfhrung in die kombinatorische Topologie. Darmstadt: WissenschaftlicheBuchgesellschaft. Check date values in: |date= (help)

9.10 References Kharlampovich, Olga; Myasnikov, Alexei (2006). Elementary theory of free non-abelian groups (PDF). J.

Algebra 302 (2): 451552. doi:10.1016/j.jalgebra.2006.03.033. MR 2293770.

W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory, Dover (1976). P.J. Higgins, 1971, Categories and Groupoids, van Nostrand, {New York}. Reprints in Theory and Appli-cations of Categories, 7 (2005) pp 1195.

Sela, Z. (2006). Diophantine geometry over groups. VI. The elementary theory of a free group.. Geom.Funct. Anal. 16 (3): 707730. MR 2238945.

J.-P. Serre, Trees, Springer (2003) (English translation of arbres, amalgames, SL2", 3rd edition, astrisque 46(1983))

9.10. REFERENCES 19

P.J. Higgins, The fundamental groupoid of a graph of groups, J. London Math. Soc. (2) {13}, (1976)145149.

Alu, Paolo (2009). Algebra: Chapter 0. AMS Bookstore. p. 70. ISBN 978-0-8218-4781-7.. Grillet, Pierre Antoine (2007). Abstract algebra. Springer. p. 27. ISBN 978-0-387-71567-4..

Chapter 10

Freiheitssatz

In mathematics, the Freiheitssatz (German: freedom/independence theorem": Freiheit + Satz) is a result in thepresentation theory of groups. The result was proposed by the German mathematician Max Dehn and proved by hisstudent, Wilhelm Magnus, in his doctoral thesis.

10.1 Statement of the theoremConsider a group presentation

G = hx1; : : : ; xnjr = 1i

given by n generators xi and a single cyclically reduced relator r. If x1 appears in r, then the subgroup of G generatedby x2, ..., xn is a free group, freely generated by x2, ..., xn. In other words, the only relations involving x2, ..., xn arethe trivial ones.

10.2 References Magnus, Wilhelm (1930). "ber diskontinuierliche Gruppen mit einer denierenden Relation. (Der Frei-heitssatz)". J. Reine Angew. Math. 163: 141165.

10.3 External links V.A. Roman'kov (2001), Freiheitssatz, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-4

20

10.4. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 21

10.4 Text and image sources, contributors, and licenses10.4.1 Text

Absolute presentation of a group Source: https://en.wikipedia.org/wiki/Absolute_presentation_of_a_group?oldid=657038606 Con-tributors: Vipul, Elonka, Dan Hoey, GurchBot, Bernard Hurley, R'n'B and Anonymous: 2

AndrewsCurtis conjecture Source: https://en.wikipedia.org/wiki/Andrews%E2%80%93Curtis_conjecture?oldid=637342779 Con-tributors: Michael Hardy, Charles Matthews, Jitse Niesen, Algebraist, Phaedriel, Akriasas, Singularity, David Eppstein, Rybu, Streaks100,JackSchmidt, Alexbot, Sabalka, Addbot, Yobot, Citation bot, Citation bot 1, ClueBot NG and Anonymous: 2

Automorphismgroup of a free group Source: https://en.wikipedia.org/wiki/Automorphism_group_of_a_free_group?oldid=446762775Contributors: Michael Hardy, R.e.b., Headbomb and Julian Birdbath

BaumslagSolitar group Source: https://en.wikipedia.org/wiki/Baumslag%E2%80%93Solitar_group?oldid=594446247 Contributors:Zundark, Michael Hardy, C S, Algebraist, Dan131m, Iain.dalton, Lhf, Jim.belk, CRGreathouse, Ntsimp, Headbomb, Turgidson, Agol,David Eppstein, Maproom, Dendrophilos, Addbot, Luckas-bot, Yobot, Rubinbot, EmausBot, ChrisGualtieri and Anonymous: 5

Combinatorial group theory Source: https://en.wikipedia.org/wiki/Combinatorial_group_theory?oldid=643205599Contributors: MichaelHardy, Charles Matthews, Cambyses, Nbarth, CBM, JohnBlackburne, JackSchmidt, Addbot, Luckas-bot, Erik9bot, HRoestBot, Rausch,Xnn, Helpful Pixie Bot, Brad7777, Qetuth, K9re11 and Anonymous: 1

Commutator collecting process Source: https://en.wikipedia.org/wiki/Commutator_collecting_process?oldid=653635876 Contribu-tors: Michael Hardy, R.e.b. and K9re11

Cyclically reducedword Source: https://en.wikipedia.org/wiki/Cyclically_reduced_word?oldid=544047721Contributors: CharlesMatthews,Mairi, Oleg Alexandrov, Algebraist, YurikBot, Jim.belk, Vanish2, Addbot, Charvest and Anonymous: 2

Fox derivative Source: https://en.wikipedia.org/wiki/Fox_derivative?oldid=663099377Contributors: CharlesMatthews, Staecker, Crasshop-per, Headbomb, Turgidson, David Eppstein, Mikemoral, Henry Delforn (old), JackSchmidt, LizardJr8, DOI bot, Ozob, FrescoBot, Cita-tion bot 1, Alpha carinae, Javert, Abc518, Trappist the monk, Monkbot and Anonymous: 2

Free group Source: https://en.wikipedia.org/wiki/Free_group?oldid=672630943 Contributors: AxelBoldt, Zundark, Tomo, MichaelHardy, Kidburla, Iorsh, Charles Matthews, Dysprosia, Robbot, Mohan ravichandran, Altenmann, MathMartin, Tobias Bergemann, Tosha,Giftlite, Dbenbenn, Sam nead, ZeroOne, Vipul, C S, Fadereu, RJFJR, Linas, Rjwilmsi, R.e.b., Marozols, Chris Pressey, Kapitolini, Lau-rentius, Archelon, Trovatore, Mikeblas, Larsbars, RonnieBrown, Reedy, Bluebot, Vina-iwbot~enwiki, Jim.belk, Mathsci, Mct mht, Cy-debot, Tiphareth, Punainen Nrtti, Dragonare82, Repliedthemockturtle, LachlanA, Turgidson, Ralamosm, David Eppstein, Robert Illes,LokiClock, TXiKiBoT, JackSchmidt, ATC2, Virginia-American, Addbot, DOI bot, , Hyginsberg, Yobot, Ziyuang, Citation bot,Omnipaedista, HenrikRueping, Citation bot 1, Foobarnix, RjwilmsiBot, Pet3ris, Helpful Pixie Bot, HUnTeR4subs, Jochen Burghardt,K9re11, Redbrave and Anonymous: 34

Freiheitssatz Source: https://en.wikipedia.org/wiki/Freiheitssatz?oldid=466500261 Contributors: Sodin, Nbarth, Cydebot, Sullivan.t.j,9258fahskh917fas, Omnipaedista, ManiacParisien and Qetuth

10.4.2 Images File:Baumslag-Solitar_Cayley.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/b2/Baumslag-Solitar_Cayley.svg Li-

cense: CC0 Contributors: Own work Original artist: Jim.belk File:Baumslag-Solitar_Cayley_3D.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/b0/Baumslag-Solitar_Cayley_3D.

svg License: CC0 Contributors: Own work Original artist: Jim.belk File:Encoder_Disc_(3-Bit).svg Source: https://upload.wikimedia.org/wikipedia/commons/9/9b/Encoder_Disc_%283-Bit%29.svg Li-

cense: Public domain Contributors: Replacement for Image:Encoder_disc.png Original artist: jjbeard File:F2_Cayley_Graph.png Source: https://upload.wikimedia.org/wikipedia/en/e/e8/F2_Cayley_Graph.png License: PD Contributors:

? Original artist: ? File:Free_Group_Universal.svg Source: https://upload.wikimedia.org/wikipedia/commons/c/cc/Free_Group_Universal.svg License:

Public domain Contributors: Own work Original artist: Jim.belk File:Rubik{}s_cube_v3.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/b6/Rubik%27s_cube_v3.svg License: CC-

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10.4.3 Content license Creative Commons Attribution-Share Alike 3.0

Absolute presentation of a groupFormal DefinitionExampleBackground References

AndrewsCurtis conjectureReferences

Automorphism group of a free groupPresentationReferences

BaumslagSolitar groupLinear representationNotesReferences

Combinatorial group theoryHistory References

Commutator collecting processStatementReferences

Cyclically reduced wordProperties References

Fox derivativeDefinitionApplicationsSee alsoReferences

Free groupHistory Examples Construction Universal property Facts and theoremsFree abelian group Tarskis problemsSee alsoNotesReferences

FreiheitssatzStatement of the theoremReferencesExternal linksText and image sources, contributors, and licensesTextImagesContent license