Appendix I - theory I. EqUivalence relation 2. Groups and a bit of ring theory 3. 4. Presentations of groups 5. problem 6. Reidemeister-Schreier method, presentation of a subgroup 7. Triangle groups

In order to build and enhance mathematical theories, it is usually necessary to borrow and apply concepts from various branches of mathematics. This book is not different in this regard. However, it would be rather cumbersome, even nowadays electronically, for the reader to keep on popping off to the library to find an explanation of some bit of or topology. So, in the first three appendices, we have gathered together the concepts and ideas that should make this book, on the whole, self-contained. In this appendix, besides providing a quick introduction to the fundamental concepts of group theory, we also cover, in some detail, less familiar group the• ory, namely, the free group, the presentation of a group, and the Reidemeister• Schreier method. Even though some of these concepts might be known to the reader, hopefully, the setting and their use in the theory of braids might shine a new perspective on these concepts .

• Equivalence relation "Equivalence?" Throughout this book we say braids are equivalent, equal and even on occasion the same. To understand completely the mathematical rigour of equivalence, we need to precisely comprehend, how one object relates to another. Hence, we need to introduce the fundamental concept of an equivalence relation. Let us begin rather abstractly, suppose "E is a non-empty set and there exists among the elements of "E some way of relating one to another.

EXAMPLE 1.1 (1) Let "E be the set of all real numbers, namely "E = lR.. We say that an element a of "E is related to another element b, if a is larger than b, or symbolically a > b. (2) Let us suppose "E is the set of all , so "E = ~. On this occasion, let us choose an arbitrary non-zero , n say. We can introduce a relation on "E as follows, an integer a is related to another integer b if a - b is divisible by n. In such circumstances, it is usual to say a is congruent to b modulo n, or in terms of mathematical notation a == b (mod n). Clearly, the above relations are not unique for their respective sets "E. Again, let us be a bit more abstract and define the notation a rv b if a is related to b 219 220 ApPENDIX I - GROUP THEORY in E. In the first example, a --- b if a > b, while in the second example, a --- b if a - b = nk for some integer k.

DEFINITION 1.1 A relation on a set E is said to be an equivalence relation if the Ifollowing 3 conditions are satisfied: (1) (Reflective) a is related to itself, Le., a --- a. (2) (Symmetrie) If a --- b then b --- a. (3) (Transitive) If a --- band b --- c then a --- c.

ExAMPLE 1.1 (1) (continued) Clearly, a > a cannot hold. Therefore, the relation given by a > b cannot be an equivalence relation. (2) (continued) We leave it as a short exercise for the reader to show that congru• ence is an equivalence relation. The notion of an equivalence relation may be thought of as the generalization of the mathematieal relation "is equal to." Hence, it should be understandable, why we may blur on occasion the distinction between equal and equivalence. Now, suppose we are given a set E with an equivalence relation. If we fix an element a in E (a E E), then we can create a subset of E by just taking all the elements that are equivalent to a. Such a subset is called an equivalence class represented by a and denoted by [al. In terms of the notation, [al consists of all the elements xE E such that a rv X.

EXAMPLE 1.1 (2) (continued) Let us fix n = 5. Then, [0] is the set {O, ±5, ±1O, ... }, [3] is the set { ... ,-7, -2,3,8 ... }.

PRoPOSITION 1.1 Suppose E is a set with an equivalence relation. If [al n [bI =10, then [al = [bI. The significance of the above proposition is that we may completely decompose E into a collection of (disjoint ) equivalence classes. We shall denote this collection by E/rv •

EXAMPLE 1.1 (2) (continued) Again, fix n = 5. Then E/rv has only 5 elements,

E/rv = {[O], [1], [2], [3], [4]}. (1.1)

Note that, [0] = [5] = [10] = ... , et cetera.

EXERCISE 1.1 Let E be a set and suppose we can decompose E into disjoint subsets Al, A2 , ••• (the number of such subsets need not be finite). For any x, y E E, let us define a relation in E as follows, x rv y if x and y both belong to the same set, Ai say. Show that the above relation is an equivalence relation and

(1.2) §2. GROUPS AND A BIT OF RING THEORY 221

As a final note to this section, suppose we have an equivalence relation on a set I:, then if a is related to b we have by definition that b is related to a. Hence, we may say without ambiguity that a and bare related.

11 Groups and a bit of ring theory IDEFINITION 2.1 A non-empty (finite or infinite) set, G, is said to be a group if the set G satisfies the following conditions: (1) To each (ordered) pair (a, b), consisting of elements a and b in G, we can assign a unique element c in G. The assignment of c to an ordered pair (a, b) is usually called a binary operation. For simplicity, c is usually called the product of a and b, and denoted by a 0 b or just by ab. (2) The product is associative, symbolically, (a 0 b) 0 c = a 0 (b 0 c). (3) There exists an element e in G such that for anyelement xE G, xoe = x and e 0 x = x. Such an element e is called the identity (element) of G. We shall usually denote e by 1. ( 4) For each element x in G, there is an element y in G such that x 0 y = e and y 0 x = e. Under these circumstances, y is said to be the inverse of x and denoted by X-I.

EXERCISE 2.1 (1) Show that even though it was not explicitly stated the identity, e, and the inverse element are both unique. (2) Show that, for x and y in a group G, (xy)-1 = y-lx -l and (x-1)-1 = X. Even a cursory glance immediately indicates that the essence of a group is the binary operationjproduct. However, a set G need not have just one way of assigning a product. So, a set with two possible products gives rise to 2 groups. If such a case arises, we will explicitly delineate the 2 products. For the most part, if the product is readily understood, we will not differentiate in our notation between the set G and the group G.

EXAMPLE 2.1 (1) The set of all integers, 2, forms a group with the binary opera• tion given by the ordinary addition. Similarly, the rational numbers, Q, the real numbers, lR, and the complex numbers, (C, are also groups. (2) The set of all non-zero rational numbers, QX, non-zero real numbers, lR x , and non-zero complex numbers, (Cx, is each a group with binary operationjproduct given by the usual multiplication. However, the set of non-zero integers does not form a group with multiplication as the product, for it is impossible to find an inverse for any integer besides l.

EXAMPLE 2.2 Let G be the set of all permutations on n numbers 1,2, ... ,n, with n ~ 2. It is standard to denote an element, (7, of G as

(2.1)

where the notation indicates that j permutes to the integer i j . 222 ApPENDIX I - GROUP THEORY

Suppose T is another element of G, Le.,

(2.2)

Clearly, we may rearrange the columns of T so that

(2.3)

Hence, we may define a product on G as

~) . (2.4)

Caveat leetor, The above product is performed from left to right. However, there is nothing in our definition of a binary operation that precludes us from starting with T and then rearranging 0'. In this case, the product would be from right to left. Such a distinction can at times be very important. The set G with the above product forms a group, in which the identity element is just the identity permutation, obtained by setting ij = j in (2.1), while the inverse of 0', denoted by 0'-1, is

(2.5)

This group G is more commonly known as the symmetrie group on n letters and denoted by Sn. While the notation above for apermutation is explicit in its clarity, at times we will employ a shorthand version of the above notation. Consider the row of r integers (al a2 ... ar ), this may be thought of a representing apermutation with ai permutating to ai+l for i = 1,2, ... ,r -1 and ar permutes to al' Such a row representation of apermutation is usually called a eycle. The product of cycles is just a matter of following, moving from left to right, the permutation of ai, say, along each consecutive cycle, Le., in the first cycle ai permutes to ai+l, in the second cycle ai+l permutes to ah' in the third cycle ah permutes to ajs' ... , in the final cycle ajk_l permutes ajk' Therefore, in the product of cycles ai permutes to ajk' If an integer m is not in a cycle, then within that cycle m permutes to itself. Note that the identity permutation in Sn consists of n cycles, (1)(2) ... (n), and we shall denote it simply by (1). EXERCISE 2.2 (1) Show that, in S4, (12)(13)(14) = (1234) and (12)(234)(12) = (134)(2) = (134). (2) Show that S~ has n! distinct elements. The total number of distinct elements of a group G is called the order of G and denoted by IGI. If IGI is finite then we say G is a finite group, otherwise, G is an infinite group. It follows immediately from Exercise 2.2(2) that Sn is a finite group. On the other hand, all the groups in Example 2.1 are infinite groups. §2. GROUPS AND A BIT OF RING THEORY 223

IDEFINITION 2.2 Let G be a group. Suppose, for any 2 elements a and b in G,

ab = ba. (2.6)

Then, G is said to be a commutative group or, equivalently, an abelian group. I A group for which (2.6) does not hold is said to be non-commutative or, equiv• alently, non-abelian .

• Note: The name abelian comes from the distinguished Norwegian mathemati• cian N .H. Abel (1802-1829) in recognition of his significant contributions to various fields of mathematics related to this concept. EXERCISE 2.3 Show that all the groups in Example 2.1 are abelian, but for n;:: 3 Sn is non-abelian. Let G be a group and suppose H is a subset of G. It is quite plausible that the product in G need not restrict to a product in H. In other words, there may exist elements a and b which are in H, and hence in G, but even though ab is in G this productjelement need not be in H. IDEFINITION 2.3 Let H be a subset of a group G. Then, H is called a subgroup of G if H inherits a group structure from G, or, equivalently, the following conditions are satisfied: (1) The binary operation in G is also a binary operation in H, Le., if a and b belong to H then ab is also in H. (2) H contains the identity element, e, of G. (3) For every element a in H, the inverse a-1 (defined initially as an element of G) also belongs to H .

• Note: Since associativity holds in G it is easy to see that it also holds on restriction to H. EXAMPLE 2.3 In Example 2.1(1), the group LZ is a subgroup of Q and also of lR. However, even though QX is a subset of Q, QX is not a subgroup of Q. An arbitrary group, G, has at least one element, e, that commutes with every element of G. In fact, the set of all such elements forms a subgroup of G called the centre of G and denoted by C(G). To be precise,

C(G) = {x E G I xy = yx for all y E G}.

EXAMPLE 2.4 Consider the group S3, see Example 2.2. Then, H = {(I), (12)} and N = {(I), (123), (132)} are both subgroups of S3. Let us investigate Sn more closely. An element of Sn is called an even (odd) permutation if it is the product of an even (odd) number of cycles of the form (ij). It is a standard fact that any permutation is either an even or odd permutation. For example, since (123) = (12)(13), (123) is an even permutation. On the other hand, by Exercise 2.2(1), (1234) is an odd permutation. 224 ApPENDIX I - GROUP THEORY

It can be shown that the set consisting of all even permutations on n letters is a subgroup of Sn, commonly known as the alternating group and denoted by An. EXERCISE 2.4 Show that a subset H of a group G is a subgroup (of G) if and only if (1) ab in H for anyelements a and b in H; (2) a-1 in H for anyelement a in H. Let H be a subgroup of a group G. Further, suppose that x is an element of G. Define the set Hx to be the set consisting of all elements hx where his in H. In mathematical notation,

Hx = {hx I h EH}. (2.7)

The set Hx need not be a subgroup of G, in fact this only occurs if x is an element of H (then Hx = H). Usually, Hx is said to be a right coset of Hin G. Similarly, if we define xH as

xH = {xh I hE H}, (2.8) then xH is called a left coset of HinG. If G is an abelian group and H is an arbitrary subgroup of G then for every x in G we have Hx = xH. EXAMPLE 2.5 Let G = S3 and as in Example 2.4 let H = {(I), (12)}. Then, the right cosets of H in Gare

H(l) = {(I), (12)} = H(12), H(13) = {(13), (123)} = H(123), H(23) = {(23), (132)} = H(132).

On the other hand, the left cosets of H in Gare

(l)H = {(I), (12)} = (12)H, (13)H = {(13), (132)} = (132)H, (23)H = {(23), (123)} = (123)H.

Therefore, we can see immediately that (13)H i= H(13). So, if G is non-abelian then for a fixed x not in H, it is quite possible that xHnHx =x.

DEFINITION 2.4 A subgroup H of a group G is called anormal subgroup if for any x in G the sets Hx and xH consist of exactly the same elements, Le., Hx = xH.

Clearly, if G is abelian then every subgroup is anormal subgroup. Providing a group G has more than 1 element, G, whether abelian or not, has at least 2 normal subgroups, Le., G and {e}. If the only normal subgroups of G are these 2 subgroups, then G is said to be simple. §2. GROUPS AND A BIT OF RING THEORY 225

The number of distinct right cosets of H in G is called the index of H in G and denoted by [G : H]. EXERCISE 2.5 (1) Show that the number of distinct right cosets of HinG is equal to the number of distinct left cosets of H in G. Hence, there is no ambiguity in our definition of the index [G : H]. (2) Show that each right coset (and hence each left coset) of H in G has the same number of elements. More precisely, show that there is a 1-1 correspondence between 2 distinct right cosets (and left cosets) of H in G. (Hint, Consider the map

EXAMPLE 2.6 (1) H in Example 2.5 is not anormal subgroup, and [G: H] = 3. (2) The alternating group, An, for n ~ 2, is anormal subgroup of Sn. In addition, there are only 2 right(1eft) cosets of An, namely, An itself and An (I2). Hence, [Sn: An] = 2, and so IAnl = n!/2' (3) The centre of a group G, C(G), is always anormal subgroup. (4) [C:~] = 2 and ~ : Q] is infinite. IDEFINmON 2.5 Let G be a group and let 9 = {gI, g2, ... } be a (finite or infinite) subset in G. Then, the smallest subgroup of G that contains 9 is called the subgroup generated by the set g. While, the smallest normal subgroup of G that contains 9 is called the normal closure of 9 in G. Further, the order 0/ an element 9 E G, denoted by Igl, is the order of the subgroup generated by {g}, or, equivalently, Igl is the smallest positive integer n I. such that gn = 1. It is easy to show the following. PROPOSmON 2.1 Let H be the subgroup oE G generated by 9 = {gI, g2, ... } and N be the normal c10sure oE g. Then, (1) Every element oE H is the finite product oE the gi or g;1 in g. Namely, H is the set oE all elements (in G) oE the Eorm

g~lg~2 ... g~k, 31 J2 Jk

where git' ... ,gik are in 9 and Ci = ±1. (Note that k is not fixed and we may trunk oE the identity element e 8S the empty product or gig; 1 . ) (2) Every element oE N is the finite product oE the conjugates oE gi or g;1 in g. Namely, N is the set oE a1l elements (in G) oE the Eorm

e1 -1)( e2 -1) ( ek -1) ( U1gi1 U1 u2gh U2 . .. Ukgjk Uk '

where git' ... ,gj" are in g, Ci = ±I and Ui are elements oE G. (Again, k is not fixed.) 226 ApPENDIX I - GROUP THEORY

IDEFINInON 2.6 Let G be a group and suppose x and y are 2 arbitrary elements of G. Then, the element xyx-ly-l (in G) is called the commutator of x and y, and denoted by [x, Yl. ! The subgroup of G generated by all commutators is called the commutator subgroup of G and usually denoted byeither G' or [G, Gl.

By Proposition 2.1, every element of G' is a product of finitely many commu• tators and their inverses, Le.,

(2.10)

where, for i = 1,2, ... k, ai, bi are elements of G, Ci = ±1 and k is not fixed. Starting with anormal subgroup N of a group G, we know that we can de• compose G into a set of mutually disjoint right cosets, M = {Nu!, NU2, ... } say. Suppose Nx and Ny are 2 arbitrary elements from M, and define

Nx °Ny = N(x oy), (2.11)

where x 0 y is just the product of elements in G. Since N is anormal subgroup of G, it can be shown that the coset N z, where z = x 0 y, is defined uniquely by (2.11). Hence, (2.11) gives us a product in M. The above product turns M into a group. In fact, Ne (= N) is the identity el• ement ofthis group, where e is the identity element of G, and Nx-l is the inverse of Nx. This group M is commonly known as the or, equivalently, factor group of G by (the normal subgroup) N, and denoted by GIN. Further, the order of GIN is exactly [G : N]. The advantage of finding and then studying a quotient group of a group G, which we are investigating, is that the quotient group, since it is a "smaller" group, more than likely has a simpler and so more readily understandable structure. Therefore, by studying the quotient group we may be able to better understand the structure of the group G itself. EXERCISE 2.6 Let G be a group, then show that: (1) For arbitrary a,b and ein G e[a,ble-l = [cae-l,ebe-ll. (2) G' is anormal subgroup of G. (3) G /G' is an abelian group.

Suppose, now, that we are given 2 groups G l and G2 and we would like to compare them. Since more than likely their respective products are different, a direct comparison may not be feasible. What we require is some way of relating the product structure on G l to the product structure on G2. !DEFINInON 2.7 Let (Gl,o) and (G2,*) be 2 groups,where °and * refertothe respective products in the 2 groups. By a mapping ! from G l to G2, denoted by ! : G l ~ G2, we mean that to each element Xl E G l we assign a unique element X2 E G2, say. The standard notation for this is !(Xl) = X2. §2. GROUPS AND A BIT OF RING THEORY 227

The mapping J is said to be a homomorphism if it satisfies the following eon• dition: For any 2 elements x and y in Gi,

J(x 0 y) = f(x) * f(y)· (2.12)

Essentially, a homomorphism is a mapping that preserves the group strueture of Gi.

IDEFINITION 2.8 A homomorphism J from the group Gi to the group G 2 is ealled an isomorphism if f is onto and one-to-one. A group Gi is said to be isomorphie to a group G 2 , and denoted by Gi ~ G 2 , if there exists an isomorphism from Gi to G 2 .

EXERCISE 2.7 Show that the relation "is isomorphie to" is an equivalenee relation. PROPOSITION 2.2 Let f be a homomorphism from a group Gi to a group G 2 . Then, (1) f(ei) = e2, where ei, for i = 1 and 2, is the identity element of Gi' (2) f(x- i ) = f(x)-l, for any x E Gi.

Using Proposition 2.2 we ean prove the following proposition. PROPOSITION 2.3 Let f be a homomorphism from a group Gi to a group G 2 . Then,

(1) The elements of Gi that map to the identity element, e2, of G 2 form a subgroup of Gi ealled the of fand denoted by K er f. Further• more, Kerf is anormal subgroup of GI. (2) Under f the elements of Gi map to a subset of G 2 , this subset is a subgroup of G 2 and denoted by f(Gi ). (3) The quotient group Gi/Kerf and f(G1 ) are isomorphie. This isomor• phism is indueed by J, in fact, the isomorphism,

EXAMPLE 2.7 We know that ]R is an abelian group, see Example 2.1(1). The set of positive real numbers, ]R+, is also an abelian group with the product given by the standard multiplieation. If we define a mapping f : ]R ---+]R+ as folIows,

(2.13) then this mapping is an isomorphism. (We leave the proof of this as an exercise for the reader. Hint, Consider logarithms.) Therefore, ]R is isomorphie to ]R+. Ifwe apply an isomorphism to a group G, then we know that its group strueture will be preserved by the mapping. In partieular, the order of G and that of its isomorphie group are equal. Therefore, for a11 intents and purposes, the 2 groups may be eonsidered to be one and the same. On many oeeasions, given a group we are investigating, it is more judieious to spent time trying to find an isomorphism to a group whose structure we know 228 ApPENDIX I - GROUP THEORY rather than directly investigating the group. Throughout the main text of this book, this is an approach we often apply.

EXERCISE 2.8 Show that the abelian groups fZ and r(Y, see Example 2.1, are not isomorphie. (Caveat lector, It is possible to find a 1-1 correspondence between fZ and Q!x, so their orders are equal. But, the problem here is to show that there is no homomorphism from fZ to Q!x that is onto and one-to-one.)

EXAMPLE 2.8 Let us fix a positive integer n, say, and let H be the set of all integer multiples of n. Then, H is a subgroup of the abelian group fZ and [fZ : H] = n. The quotient group fZ/H is usually called the group 01 integers modulo n, and denoted by fZ/(n) or, equivalently, by fZn. This is the definition, but how can we represent the elements of fZ / (n) ? Each coset is of the form Hx = {x + nk, k = 0, ±1, ±2, ... }. If we hark back to Example 1.1(2) and in partieular (1.1), it is easy to see that

fZ/(n) = {[O], [1], ... , [n - I]}, (2.14) where [i] = { ... i - n, i, i + n, i + 2n, ... } for i = 1,2, '" , n - 1, and addition is given by the rule [i] + [j] = [k], where k is obtained from

i + j == k (mod n) (2.15) with 0 :::; k :::; n - 1. The notation in (2.14) is a bit cumbersome, clearly, there is an obvious isomor• phism between the group in (2.14) and the group

G = {O, 1,2, ... ,n -I} (2.16) where the binary operation in G is given by (2.15). So, as is more usual, we think of fZ / (n) as the finite group consisting of n integers 0,1, ... , n - 1 and the binary operation given by (2.15).

EXAMPLE 2.9 Let p be a prime number and fZ/(p) x be the set

fZ/(p) X ={1,2, ... ,p-l}. (2.17)

Then, we can make fZ/(p) x into a group by defining the product as folIows. Suppose a, b are in fZ/(p) x, then ab = c with ab == c (mod p) and 1 :::; c:::; p-1.

EXERCISE 2.9 (1) Show that the choiee of c, above, is unique, Le., there does not exist a c' (=J c) such that ab == c' (mod p) and 1 :::; c' :::; p - 1. (2) Show that for the group fZ/(p) x the identity element is 1. (3) Show that for any integer a with 1 :::; a :::; p - 1 there exists an integer b, again with 1 :::; b :::; p - 1, such that ab == 1 (mod p). (Caveat lector, If p is not prime then this part may not be true.) §2. GROUPS AND A BIT OF RING THEORY 229

EXERCISE 2.10 Show that 7l./(7) x is isomorphie to 7l./(6)' (In general, it can be shown that 7l. / (p) x is isomorphie to 7l. / (p _ 1) for any prime p.) So far we have discussed isomorphisms between two different groups GI and G 2 . What happens if we set GI = G = G 2? Clearly, the identity mapping is an isomorphism, and if G is abelian then the mapping that sends x (in G) to X-I (also in G) is an isomorphism. An isomorphism of G to itself is called an automorphism, and the set of all automorphisms of G is denoted by Aut(G). In fact, if we define a product on Aut(G) as the standard composition of two mappings, then Aut(G) becomes a group. To be consistent with notation, if land 9 are in Aut(G) then in the product Ig we first apply the automorphism land then g. However, it is possible to define the product such that we perform 9 first and then f. Suppose x is a fixed element of G and suppose y is an arbitrary element of G. Define a mapping, Ix : G -----t G, as follows, Ix(Y) = x-Iyx. (2.18) Then, Ix is an automorphism of G, and is usually referred to as an inner automorphism of G (induced by x). The set of all inner automorphisms, Ix (induced by x E G), forms anormal subgroup of Aut(G) and is denoted by Inn(G). In passing, we should note that lulv = luv.

PROPOSITION 2.4 Let C(G) be the centre oE G Then, Inn(G) ~ G /C(G)' (2.19) ThereEore, iE C(G) = {e} then G ~ Inn(G).

As an adjunct to group theory, let us momentarily turn our intention to the concept of rings. However, since we do not use rings extensively, we will not delve too deeply, just outlining the concepts that are necessary for the main text. IDEFINITION 2.9 Let R be a non-empty set. Suppose in R there are 2 binary operations, whieh we shall denote by + and 0 and call, respectively, addition and multiplication, and they satisfy the following conditions: (1) R is an abelian group with a binary operation given by addition +, the identity element is denoted by 0, and the inverse of an element a is denoted by -a. (2) Multiplication in R is associative, i.e.,

(a 0 b) 0 c = a 0 (b 0 c). (2.20) (3) The distributive law holds for the 2 binary operations, namely,

a 0 (b + c) = a 0 b + a 0 c, (2.21 )

(b + c) 0 a = boa + co a. (2.22) If the above 3 conditions hold for R, then R is said to be a ring. 230 ApPENDIX I - GROUP THEORY

Further , if, for any a and b in R, we have a 0 b = boa, then R is said to be a commutative ring. If there exists an element, 1 say, such that 10 a = a and a 01 = a for any a in R, then 1 is called a multiplicative identity and R is said to be a ring with an identity. Finally, an element u for which we can find an element v such that u 0 v = 1 Iand v 0 u = 1 is called a unit of R. Also, v is called the multiplicative inverse of u and is denoted by u -1.

EXAMPLE 2.10 (1) The set of integers, :2::, is a (commutative) ring under the standard binary operations of addition and multiplication. The ring :2:: has 2 units, the elements 1 and -1. (2) The sets Q, ~ and C all become (commutative) rings with an identity under the usual addition and multiplication. In addition, in these rings, every non-zero element is a unit, such commutative rings are commonly called fields. However, besides this aside, we will not discuss the exact nature or properties of a field. (3) :2::/(n) , see Example 2.8, is a commutative ring with an identity for every positive integer n ~ 2. If n is also prime, p say, then it can be shown that every non-zero element of :2::/(p) is a unit (see Example 2.9). Hence, :2::/(p) is a field.

EXAMPLE 2.11 Let :2::[t, r 1] be the set of all integer polynomials in the indeter• minant t with the possibility that some of the exponents may be negative, such polynomials are usually called Laurent polynomials. Namely,

where k and l are finite but not fixed and ai E:2:: with a-k # 0 # al. With the standard not ion of addition, we can find for :2::[t, Cl] an identity 0 and inverse -p(t) for each element p(t). In addition, if we consider the standard way of multiplying two polynomials, then :2::[t, Cl] becomes a commutative ring. Since p( t) = 1 is an element of this ring, :2::[t, t-1 ] is a commutative ring with an identity. Finally, it is easy to see that the units of :2::[t, r 1] are the monomials tr for every integer r. The ring :2::[t, t-1] is often referred to as the ring 0/ Laurent polynomials. Simi• larly, we can define the ring of Laurent polynomials in 0 with integer coefficients, where we set (0)2 = t. This ring is denoted by :2::[0, Jt].

EXAMPLE 2.12 Let R be a commutative ring, the previous examples give us a fair number of possibilities. Then, the set of all n x n matrices for n ~ 1 and with entries in R forms a non-commutative ring under the standard matrix addition and multiplication. This ring is usually called a matrix ring and denoted by M(n, R). In a matrix ring, if a matrix A is a unit then its determinant is non-zero. However, the converse of this statement need not be true. Importantly for our discussions in the main text, a homomorphism / from a group G to M(n, R), for some n, is called a matrix representation of G. We §3. FREE GROUP 231 leave the term representation of G to a homomorphism from G to the symmetrie group Sn, again for some n. EXAMPLE 2.13 Let, for the time being, i, j and k be abstract symbols, and consider the set, !BI, of all elements of the form

a + bi + cj + dk, where a,b,c and d are real numbers. Now, suppose that i,j and k are subject to the following formulae: (1) i 2 =j2 = k2 = -1; (2) ij = k, jk = i, ki = j; (3) ji = -k, kj = -i, ik = -j. With the above in mind, for arbitrary elements A = a1 + b1i + cd + d1k and B = a2 + b2i + c2j + d2k of !BI, define an addition and multiplication as follows,

(2.24) AB = (a1a2 - b1b2 - C1C2 - d1d2) + (a 1b2 + a2b1 + c1d2 - C2dt}i + (a1c2 + a2C1 - b1d2 + b2dt}j + (a 1d2 + a2d1 + b1C2 - b2C1)k (2.25)

One way round the need to memorize (2.24) and (2.25) is to think of A and B as polynomials in variables i,j and k. Then, we can add and multiply these "poly• nomials" in the usual fashion for polynomials, but the order of the multiplication should be carefully observed, namely, ij and ji should be clearly differentiated in this expression. Finally, apply the identities in (1), (2) and (3) above. The above addition and multiplication make lHI into a ring, often called the quaternion ring and the elements are called quaternions. In asense, the quaternions may be thought of as a natural extension of the complex numbers. EXERCISE 2.11 (1) Show that

(1 + 2i - 3j + k)(2 - i + 3j - 2k) = 15 - 6i + Oj - 9k. (2.26)

(2) Show that for any real number x, y, z and u,

(x + yi + zj + uk)(x - yi - zj - uk) = x2 + y2 + Z2 + u2. (2.27)

Using this (or otherwise), find the inverse of 2 + i - 3j + k . • Note: Since every non-zero element of lHI is a unit, lHI is a "non-commutative" field. Such a non-commutative ring is called a skew jield.

11 Free group Let 8 = {XbX2, ••. ,xn } be a set of n distinct abstract elements. Although 8 can be an infinite set, we shall assume, for simplicity, that 8 is a finite set. Similarly, let 8-1 = {X 11,X21, .. , ,X;;l} be another set of n distinct elements. 232 ApPENDIX I - GROUP THEORY

We define a ward W with elements from S U S-1 as

(3.1) where 1 ~ i b i 2 , ... ,ik ~ n and Ci = +1 or -1. As is the convention, we write Xi instead of XiI. Due to (3.1), a word W can be said to be a finite product of (the 2n) elements in S U S-I. The integer k is usually called the length of the word W As is the custom, for any natural number l we denote

-1 -1 -I Xi ... Xi - Xi 1 and Xi ... Xi = Xi . (3.2) '--" '--.-" 1 times 1 times

Let W be the set of all words formed from the elements of SuS-1. In addition, we assurne that W contains a further special element, the "empty" word, called the identity and denoted by 1. Our aim is to make W into a group, for this to happen we must define a binary operation, which we shall term the product 0/ words. el ep d 1/1 7Jq 2 b't d . W Suppose W 1 = XiI ... xip an w:2 = x j1 ..• X jq are ar 1 rary wor sm, with 1 ~ i b ... ,ip, jl, ... ,jq ~ n and cik' 1}jz are equal to either +1 or -1. We define the product of words W1 and W2 , and denoted by W1 W2 , as

(3.3)

In particular, define

(3.4)

It follows immediately from (3.3) that the product of words is associative. How• ever, even though from (3.4) we have an identity, at present we have no formal definition of an inverse of a word. Hence, as yet W cannot be made into a group. Therefore, to construct a group from W, we shall introduce the concept of equivalence between 2 words W1 and W2 .

IDEFINITION 3.1 We say a word W1 is equivalent to a word W2 if we may obtain W2 by applying a finite number of times the following two operations to W1 :

Tl' q ej ej+l e", el ejl e el 0j+l e", . XiI ... Xij X ij+1 ••• Xi", ---t XiI ... Xij XpXp X ij+l ... Xi", ' (3.5) where C = ±1, and T2' el ej e -e ej+ 1 0", el ej ej+l e", . XiI ... Xij IXpX p IX i j+l ... Xi", ---t XiI ... Xij X i j+l ... Xi", . (3.6)

IThe operation Tl is termed an insertion and T2 is termed adeletion.

EXAMPLE 3.1 Let us set S = {a, b, c} and S-1 = {a-l, b-l, c-1}. Then, the word W1 = bab- 1cc- 1ba is equivalent to ba2 . §3. FREE GROUP 233

EXERCISE 3.1 Show that the above not ion of equivalence is consistent with our definition of equivalence, Definition 1.l. As a result of Exercise 3.1, we can unambiguously say that words Wl and W2 , obtained by means of Definition 3.1, are equivalent, and we shall denote this equivalence by W 1 '" W 2. Let W be the set of all equivalence dasses, [x], formed from W with the above definition of equivalence. Namely,

[xl = {y E W I y '" x}. (3.7)

THEOREM 3.1 W is a group witb tbe produet inberited from tbe produet on W, i.e., [xl 0 [y] = [x 0 y]. Tbe identity element is given by [1]. Wbile, tbe inverse, [xt 1, of [xl is given a,s follows, if tben (3.8)

W is called the free group freely generated by xl, X2, ... , xn , or just simply a free group. The integer n is called the rank of the free group. In fact, a free group is completely determined by its rank. THEOREM 3.2 Two free groups of tbe same mnk are isomorpbie and tbe eonverse also bo1ds.

Usually, a free group freely generated by the elements Xl, x2, ... , Xn is denoted by F(Xl,X2, ... ,Xn ). Strictly speaking, an element in F(Xl,X2, ... ,xn ) is an equivalence dass of words, however, by abuse of notation, it is common for an element tü be taken as a würd from its equivalence dass. Für example, returning to Example 3.1, we say a,ab are elements of F(a,b,c). A word W (or element) of F(Xl, X2, ... , xn ) is said to be reduced if W is the word with the shortest length that represents the element W. (Equivalently, we can say a word is reduced if it is not possible to apply any furt her T2 to the word.) Again returning to Example 3.1, abb-1e is not a reduced word, but ac and aba-1 are reduced words. Since the following proposition is easily proven by induction on the length of a word, we leave it as an exercise. PROPOSITION 3.3 Let 9 be an element of F(Xl,X2, ... ,xn ). If 9 eommutes witb xq , tben 9 = x~ for some integer k. More genemlly, if words W 1 and W2 eommute, tben eitber W2 = Wf or W1 = W~, for some integers k and l. EXERCISE 3.2 (1) Show that a free group, F, of rank n, is abelian if and only if n=1. (2) Show that the centre of a free group F of rank n > 1 is a trivial group. 234 ApPENDIX I - GROUP THEORY

• Presentations of groups Let G be a group. Then G has a (finite or infinite) set of elements 9 = {gb g2, ... } that generate G. In other words, every (non-identity) element 9 of G can be expressed as a finite product of the elements in 9 and their inverses, (4.1) where 1 ~ ib i2, ... ,ik and Cj is either +1 or -l. The set 9 is called the set 0/ generators of G, and we say that 9 gene rates the group G. For any group such a set will always exist, to find one such set just take alt the elements of G. EXAMPLE 4.1 (1) The set {I} is the (set of) generator of the additive group Z, since every integer m is the sum of Iml copies of 1 or -l. (2) Let G = 83 , Then, 9 = {(12), (13), (23)} (4.2) is a set of generators of G. We leave it as an exercise to show that, in fact, 9 is a set of generators, and further the sets g' = {(12), (13)} and g" = {(12), (123)} are also sets of generators of 83 . Now, let 9 = {gb 92, ... } be a set of generators of a given group G. Cor• responding to g, we can construct a free group F freely generated by the set X = {Xl. X2, ... } that is in 1-1 correspondence with g. PROPOSmON 4.1 The mapping / : X ----+ 9 given by !(Xi) = 9i for i = 1,2, ... , can be extended to a (onto) homomorphism 1: F ----+ G. PROOF Let y = x~: x~: ... x~: be an element of F. We leave it to the reader to show that the mapping 1: F ----+ G given by j(y) = g~llgf: ... 9f: (4.3) is the required mapping. (In essence, what needs to be shown is that the mapping defined by (4.3) is well-defined.) o By Propositions 2.3(3) and 4.1, we have an isomorphism,

(X I R) = (Xt,X2, •.. I Rb R2 , ..• ), (4.4) which we call a presentation of the group G. §4. PRESENTATIONS OF GROUPS 235

On occasion, we shall write the right-hand side of (4.4) as

(Xl,X2,···IRl =I,R2 =1, .. ·) or (4.5)

to emphasize that each Ri belongs to the kernel and j(Ri ) is the identity element in G. Each R i is called a relator and R i = 1 is called a relation or a defining relation of G. If an element 9 in F is a product of conjugates of relators R i or Ri 1 in R, then 9 belongs to N and is called a consequence of the relators (or relations) Rb R2, ... (or Rl = 1, R2 = 1, ... ). Intuitively, the group G may be thought of as the group generated by Xl, X2, ... in which the equalities Rl = 1, R2 = 1, ... hold. Also, sometimes, the relation R = 1 will be written as Wl = W2 where R = Wl W2- l . For instance, a relation Xl X2X 11 x2l = 1 can be rewritten as Xl X2 = X2Xl. This latter notation, is probably more comfortable/familiar. Often a presentation is identified with the group, and we write

(4.6)

If both sets X and R are finite sets, then the presentation (4.6) is said to be a finite presentation or, equivalently, G is finitely presented. In general, to find a presentation for a given group is quite an arduous task. At the beginning of the 20th century, however, this task was of partieular interest when an exhaustive search was made to associate groups with various geometrie structures. For the case of a free group G of rank n, a presentation of G is quite easy to write down, (4.7) where -- indieates that there are no relations for the given group, except for trivial relations of the type xixil = 1, whieh are usually omitted . • Note: For a free group, the normal closure ofthe relators, N, is a trivial group. The trivial group consisting of only the identity element, 1, has presentations

(- I -) or (x I x) or (x I X = 1 ). (4.8)

EXAMPLE 4.2 For G = 8 3 , we see that 9 = {(12), (13), (23)} is a set of generators and hence X = {a,b,c}. (For clarity, we shall use a,b,c rather than Xl,X2,X3, respectively.) So, a, band c correspond, respectively, to (12), (13) and (23). A presentation for G = 83 is given by

(a, b, cl a2 = b2 = c2 = abac = (ac)3 = 1). (4.9) We can eliminate c from (4.9) by replacing c by a-lb-la-l in the remaining relators. So, c2 = 1 is replaced by

( a -lb-l a -1)( a -lb-1 a -1) -a_ -lb-1 a -2b-1 a.-1 (4.10) 236 ApPENDIX I - GROUP THEORY

However, using a2 = 1 (and hence a-2 = 1) and b2 = 1 (and hence b-2 = 1), (4.10) becomes the trivial relation, 1 = 1. In other words, the element (4.10) is a consequence of the two relations a2 = 1 and b2 = 1. Therefore, from our list of relations we can remove the relations c2 = 1 and abac = 1, and from our list of generators we can remove the generator c. Hence, (4.9) can be rewrittenjsimplified as follows,

(a,b I a2 = b2 = (ab)3 = 1). (4.11)

EXERCISE 4.1 Show that 8 3 also has the presentation

(X,y I x2 = y3 = (xy)2 = 1). (4.12)

(Hint, Substitute a for x and b for xy in (4.11). That the set {x,y} generates 8 3 follows since a (= x) and a-lb (= y) generate 8 3 .) The group G given by the presentation

(4.13) can be interpreted in a manner similar to a free group. Firstly, let F be the free group generated by Xl, X2, ... , x n . Then, let R = {Rl , R2 , ... , Rm } be a set of words in F. We define an equivalence relation, rv, in F (relative to R) as follows: R Given 2 words W and W' in F, we say W is equivalent to W' (relative to R,) and denoted by W rv W', if W can be transformed into W' by a finite R sequence of Tl and T2 operations and the following 2 operations, firstly,

where 1/ = ±1 and Uk is a word in F. While, the second operation T2R (deletion) is the inverse operation of T1R. We leave it as an exercise for the reader to show that the above notion of equivalence is an equivalence relation. Then, the set W of all equivalent classes (relative to R), formed from the set of all words W in F, is a group. Now, the natural homomorphism 'P : W ----> G = F jN, where N is the normal closure of {Rl, ... Rm }, defined by

(4.14) induces an isomorphism. Therefore, (4.13) is also a presentation of the the group W. This can be used to determine a presentation of a group as follows. Suppose we are given a group G and we want to show that it has the presen• tation, (Xl, ... X n I R l = 1, ... ,Rm = 1). §4. PRESENTATIONS OF GROUPS 237

Firstly, take the n elements gl, ... , gn in G and show that {gl, .. , , gn} gen• erates G. Let cp be areplacement mapping that sends gi (or g;l) to Xi (or x;l) in each word from {gl, ... gn, gl\ ... g.;;:l} Further, let 'IjJ be the inverse replacement mapping that interchanges Xi and gi in each word in {Xl, ... Xn, x l 1, ... X.;;:l}. Next, show that if 9 = 1 in G, then

and conversely we have 'IjJ(Ri ) = 1 in G for i = 1,2, ... , m. The algorithm is now complete.

EXERCISE 4.2 Let G = (x, y I x2 = y3 = (xy)5 = 1). (4.15) Show that in G (4.16)

EXERCISE 4.3 Show that the abelian group Z I (n) has a presentation

(x1Ix~=I). ( 4.17)

(Hint, a homomorphism !: F(X1) ---4 Z/(n) is given by !(X1) = 1.) • Note: The product in F is multiplicative but in Z/(n) it is additive. To end this section, we would like to mention a few theorems that will be used frequently throughout this book.

THEOREM 4.2 Let (Xl, ... X n I R I = 1, R 2 = 1, ... ,Rm = 1) be a presentation of a group G. Let {Sl,S2, ... ,SI} be a set of elements of G, where each Si is a word in Xl, X2, ... , Xn and x l 1, X21, ... ,x.;;:l. Further, let N be the normal c10sure of {51, 52, ... , SI} in G. Then, the quotient group GIN has a presentation of the form:

(Xl, ... Xn I R1 = 1, R2 = 1, ... ,Rm = 1, 51 = 1, 52 = 1, ... , SI = 1). (4.18)

It is easy to see that GIN is "simpler" than the original group G, in the respect that with more relations this presentation, in all likelihood, can be more easily simplified. With an appropriate choice of relations, often the important properties of G can be preserved in the group GIN.

EXERCISE 4.4 Let G = (a, b, cl a2b-2 = 1, b2c-2 = 1), and let N be the normal closure of c2 in G. Then,

GIN = (a,b,c I a2b-2 = 1, b2c-2 = 1, c2 = 1) = (a, b, cl a2 = 1, b2 = 1, c2 = 1). (4.19) 238 ApPENDIX I - GROUP THEORY

The following theorem will prove to be a very useful tool throughout the book. THEOREM 4.3 Let G = (Xl, ... xn I R1 = 1, R2 = 1, ... ,Rm = 1) where R j = xj:xj: ... xj~, with 1 ~ jl.h, ... jk ~ n and Ci = ±l. Further, let H be an arbitrary group and f a mapping from F( Xl. X2, ... ,xn ) to H defined by (4.20) for i = 1,2, ... ,no IE, for j = 1,2, ... ,m,

(4.21) in H, then f defines a homomorphism 1: G ---t H with

(4.22)

(For proofs, see [MKS*] or [ex*].) EXAMPLE 4.3 Let G = (a, b I a4 = b2 = (ab)3 = 1). Let us define a mapping f : F( a, b) ---t 8 4 by f(a) = (1234) and f(b) = (12). Then,

f(a4 ) = (1), f(b2 ) = (1) and f«ab)3) = (1), and hence f defines a homomorphism 1: G ---t 8 4 with [(a) = (1234) and [(b) = (12).

11 Word problem Suppose we are given a group G by means of a presentation, namely,

G = (Xl.X2, ... ,Xn I R 1 = 1, R2 = 1, ... ,Rm = 1), (5.1) with the possibility that m andjor n may be infinite. From such adefinition of a group, an arbitrary element 9 of G is expressed as a word in the generators Xi and their inverses. WORD PROBLEM Find a reasonably practical method that will be able to decide whether or not two arbitrary words 91 and 92 in G are equal, or, equivalently, given an element 9 (= 9192 1 ) is it equal to 1, the identity element in G? As might be expected, the word problem is one of the fundamental problems of group theory. Unfortunately, there is no guarantee that a method exists. However, if a method exists, then we say the word problem is solvable for G. Otherwise, the word problem is not solvable for G. §5. WORD PROBLEM 239

In the above, we say the method must be reasonably practical. But, this is an extremely vague and not very mathematical statement. In fact, to be more precise, the problem should really lie within the realm of mathematical logic rather than group theory. So the crux, of a primary stab at a solution of the word problem, is to try and make dear, mathematically, what is possible when we say "reasonably practical." A good way of doing this is to look at some groups for which the word problem is solvable.

THEOREM 5.1 The word problem for a free group is solvable.

PROOF Let F be a free group of rank n generated by Xl. X2, ... ,Xn . An element

(5.2) of F is equal to the empty word, 1, if and only if we can eliminate each x:;, bya series of Tl or T2 transformations. That is to say, we can only cancel products within 9 of the form XiX;l or X;lXi. If we cannot find such cancellations, then 9 is never equal to the empty word. Therefore, to solve the word problem for an arbitrary word, 9, of a free group, F, we need only check if XiX;l or X;lXi exists within the word 9. Such a straight• forward method can be deemed reasonably practical, so the word problem may be said to be solvable for a free group. o EXAMPLE 5.1 Let F = (a,b,c I-). In F, neither the word 91 = aba-lb nor 92 = b-la2a-lbaa-lb-2a-l is equal to the identity. However, 91 = 921, Le., 9192 = 1.

THEOREM 5.2 Let G be a group with the following presentation,

(5.3) where PI, P2, ... ,Pn are integers. Then, the word problem is solvable for G. In fact, it is known, [Nu], that every non-trivial element, 9, of the above group can be uniquely written in the following form,

(5.4) where Al, A2, ... ,Aq are integers such that 1 ~ Aj < Pi; for each j = 1,2, ... ,q and i j 1:- i j +1 1:- ij+2 for any j = 1,2, ... ,q - 2. From the above observations, 9 = 1 if and only if 9 is an empty word. Hence, 91 = 92 if and only if both are of exactly the same form. THEOREM 5.3 The word problem is solvable for any finitely generated abelian group. 240 ApPENDIX I - GROUP THEORY

So far we have dealt with groups for which the word problem is solvable. In contrast, we should also take stock of the following theorem.

THEOREM 5.4 There exists a group for which the word problem is not solvable. To prove the theorem we need to circumscribe mathematically what we mean by solvability. As noted above, this means we must turn attention to the field of mathematicallogic. For more details, we refer the reader to [Mt] .

• Reidemeister-Schreier method. presentation of asubgroup One of the most applicable theorems in group theory outlines for a given pre• sentation of a group a method, usually called the Reidemeister-Schreier method, to determine explicitly presentations for its subgroups. Since the method plays a crucial part in various sections of this book, we would like to take the time here to describe how the method works. For a proof of why the Reidemeister-Schreier method works we refer the reader to [MKS*]. Suppose we are given a presentation of group, G, which we may take as

• Note: The method can be applied equally to the finite and infinite case. How• ever, in order not to encumber ourselves with unwieldy notation, we shall assume that both the number of generators and relations are finite. Now, let H be a subgroup of G. In trying to find a presentation for H in terms of the generators and relations given in (6.1), we need to have at hand a complete set of right coset representatives of H in G, refer to Section 2. In other words, given a subgroup H of G, we should know or we are given a complete system M of right coset representatives of H in G, Le.,

(6.2) where each Mi must be explicitly written as a word in Xl, X2, ... ,Xn or their inverses. The order of M can be infinite. Furthermore, we assume that given an element 9 of G and a set of right coset representatives, there exists a method that allows us to explicitly find the right coset representative Mj which represents Hg. Since the above is the fulcrum of the theory, the reader should take some time to fully understand its meaning. To further help the reader's un~erstanding, we present the following example.

EXAMPLE 6.1 Let G be a group given by

G = (Xl, x2, ... ,Xn I X~ = X~ = ... = X~ = 1). (6.3)

Let C(G) be the centre of G. We cannot find a presentation of C(G) by the Reidemeister-Schreier method unless we are explicitly given the complete set of right coset representatives of C(G) in G. In general, this is not the case. §6. REIDEMEISTER-SCHREIER METHOD 241

On the other hand, let G' be the eommutator subgroup of G. Then, we ean find a presentation of G' in terms of Xi and xiI using this method. We know that G' is anormal subgroup of G, and so, each right eoset representative of G' in G eorresponds to an element of G IG/. Now, sinee G IG' is an abelian group, everyelement of G IG' ean be written as G' X~I X~2 ••. x~n, where 0 ~ ab a2, ... ,an< 3. Henee, the set of 3n elements

(6.4) is a eomplete set of right eoset representatives of G' in G. Further , we know precisely the right eoset representative of the eoset eontaining a given element g. Therefore, we ean find a presentation of G' by the Reidemeister• Schreier method. So what exact1y is the method? Suppose we are given a presentation of group G as in (6.1), and further let H be a subgroup of G that satisfies the eonditions stated at the beginning of this seetion. As above, let M = {MI, M2, ... } be the eomplete set of right eoset representatives of H in G. Therefore, everyelement, g, of G is written as (6.5) for some h E Hand Mi E M. In addition, we may write

(6.6) where 1 ~ iI,i2, ... ,ik ~ n and cl = ±1. We mayassume MI = 1 is the right eoset representative of H.

IDEFINmON 6.1 The set M = {MI, M2 , ••. } is said to be a Schreier system (or satisfies the Schreier condition) if for each Mi in (6.6) the following k - 1 eonseeutive, initial parts of Mi cl CI C2 cl C2 cs cl c2 Ck_1 XiI' XiI X i2 ' XiI X i2 Xis' .•. , XiI X i2 ... X ik _ 1 Ialso belong to M.

PROPOSIOON 6.1 Suppose G is a group whose presentation we know. If H is a subgroup oE G, then there exists a Schreier system oE right coset representatives oE H in G.

EXAMPLE 6.1 (eontinued) For n = 2, the order of M is 9 and the set of right eoset representatives

(6.7)

is a Schreier system. 242 ApPENDIX I - GROUP THEORY

However, the set of right coset representatives

(6.8) is not a Schreier system. For suppose it is a Schreier system, then x~ would have to belong to M', because X~X2 is in M'. Clearly, this is not the case. On the other hand, since G'Xl1 = G'X~, if we replace X~X2 by xl1x2 in M', then M' does indeed become a Schreier system. EXERCISE 6.1 Find the Schreier system for the group in (6.3) and n = 3. Now, for an element 9 in G, let g denote the right coset representative in M of the coset Hg. For instance, if 9 is in H, then g = 1.

PROPOSITION 6.2 (1) For anyelement 9 in G and Mi in M, for i = 1,2, ... ,

(6.9) is an element of H. (2) For j = 1,2, ... ,n and k = 1,2, ... ,[G : H], let us set

(6.10)

Then, H is generated by Yj,k and their inverses.

PROOF (1) Since Mig and Mig have the same right coset representative of H, ---1 it follows that MigMig is in H. (2) First note that (M -1) -1 (! k,X,J =Y'lJ, (6.11) for some l such that M k = M1Xj. Let h be an element of H, then h can be written as

(6.12)

Of course, (6.13) Now, we transform h as follows,

The transformation in (6.14) is usually referred to as the rewriting process. §6. REIDEMEISTER-SCHREIER METHOD 243

We shall often denote by r the transformation in (6.14). In other words,

( ei e2 ep ) (1 el) (~e2) ( el e2 ep-l ep ) r Xii X i2 ... X ip = e ,XiI eXil' X i2 ..• eXil X i2 ..• Xip_ l ,Xip

_ el e2 ep - Yil,AIYi2,A2 ... Yip,Ap · Therefore, in the final equation in (6.14), each factor is either Yj,k or its inverse, for some j and k. o The next step is to find the relations that hold in H. THEOREM 6.3 Let G be a group with a presentation as in (6.1) and suppose H is a subgroup of G. If M = {MI = 1, M 2 , ... } is a Schreier system, then H has the following presentation, H = (Yj,k I Rl,k for j = 1,2, ... ,n, 1 = 1,2, ... ,m, k = 1,2, ... ,[G : H]), (6.15) where (6.16)

• Note: In the above theorem, it is quite possible for [G : H] to be infinite. REMARK 6.1 The Reidemeister-Schreier method also works if instead of using right coset representatives we use left coset representatives. However, the set of gener• ators of the subgroup is slightly different from our case, for details see [MKS*]. In the case when [G : H] is infinite, H is formally generated by an infinite number of generators, and at the same time has an infinite number of relations. However, by judicious substitution and elimination, it is possible, on occasion, to whittle down the presentation to a finite number of generators and relations. In itself, the notation in (6.16) is elegant in its compactness, but, for practical purposes, we should by means of the rewriting process, see (6.14), express these relations in terms of Yj,k and their inverses. Our aim now is to precisely illustrate the method by an example EXAMPLE 6.1(continued) Let G = (a, b I a3 = b3 = 1) (6.17) and let G' be the commutator subgroup of G. A Schreier system is M = {I, a, b, a2, ab, b2, a2b, ab2, a2b2}. (6.18) If we write ap,q = e(aPbq, a) and bp,q = e(aPbq, b), then the set of generators can be divided into two parts, firstly, aoo = 1 ,aOl = ba b-l a -1 ,a02 = b2a b- 2 a -1 , -l -2 2 2 alO = 1 ,an = a b a b a ,a12 = a b a b- a -2 , a20 = a3 , a21 = a2bab- 1 , a22 = a2b2ab-2, and secondly, 244 ApPENDIX I - GROUP THEORY

boo = 1, bOl = 1, b02 = b3, bIO = 1, bll = 1, b12 = ab3a-l , b20 = 1, b2l = 1, b22 = a2b3a-2.

Therefore, there are 10 non-trivial generators,

The set of relations are

= aOOalOa20 = a20, r(aa3a-1) = e(1, a)e(a, a)e(a2, a)e(l, a)e(a, a-1) = aOOaIO a20 aOO aoo-1 = a20, r(ba3b-l ) = e(l, b)e(b, a)e( ab, a)e( a2b, a)e(b, b-l )

Similarly, we obtain, r(a2a3a-2) = a20, r(aba3(ab)-1) = alla2laOI, r(b2a3b-2) = a02 a12 a22, raa( 2b 3b- l a -2) = a2laOlall, raa( b2 3b-2 a -1) = a12a22a02, raa( 2b2 3b-2 a -2) = a22a02a12. Also, r(lb3 1-l ) = e(1,b)e(b,b)e(b2,b) = b02 , r(ab3a-l ) = e(l,a)e(a, b)e(ab, b)e(ab2,b)e(a,a-1) = b12, r(bb3b-1) = e(1,b)e(b,b)e(b2,b)e(1,b)e(b,b-l ) = b02 · Finally, r{a2b3a-2 ) = b22 ,

r(abb3ab-1a-1) = b12 , r(b2b3b-2) = b02 , r(a2bb3b-1a-2) = b22 ,

r(ab2b3b-2a-1) = b12 , r(a2b2b3b-la-2) = b22 ·

Using the relations for G', the generators b02 , b12 and b22 can be eliminated (this is to be expected since b02 = b3 and b3 = 1 in G). After considerable simplification, there are only 6 relations,

a20 = 1, aOlalla2l = 1, a02a12a22 = 1, b02 = 1, b12 = 1, b22 = 1. §6. REIDEMEISTER-SCHREIER METHOn 245

Therefore, a presentation of G' is

(6.19)

Obviously, we can eliminate a21 and a22 using the relations a21 = al} aül and a22 = a11aül, and so finally we obtain,

(6.20)

Therefore, the commutator subgroup, G' of G is a free group of rank 4 and it is freely generated by

aOl = bab-1a-1, a02 = b2ab-2a-1, an = abab-1a-2, a12 = ab2ab-2a-2.

It is easy to see that the four elements above belong to G', but it is an altogether different matter to see that they are free generators of G'. EXERCISE 6.2 Express the element a2bab-2abab-1a-1b in terms of aij' The Reidemeister-Schreier method gives us a methodical way of obtaining a presentation of subgroup H given the presentation of the group G. However, if [G : H] is large then the process can become unwieldy. So, it is useful to keep the following in mind, since they can simplify the process. (1) If a relator R is the product of only elements of H, then R = 1 is also a relation in H. (2) If a generator g(Mk' Xj) = Yjk is a relator in G, then Yjk must be an identity element. (This can be shown once we list all the relations for H, but, we may assume before that stage that Yjk = 1. Clearly, this will greatly reduce the congestion of generators and relations.) (3) Usually, in the writing process of r(MkR1Mi/), using the Schreier right coset system, the first several factors are trivial. As a conclusion to this section, we would like to note a couple of direct conse• quences of Theorem 6.3.

THEOREM 6.4 Any subgroup H oE a finitely presented group G is finitely presented itselE iE [G : H] is finite.

THEOREM 6.5 A subgroup oE a free group is a free group.

EXERCISE 6.3 Let G = (a, bl a3 = b2 = (ab)2 = 1). (6.21) Further , let H be a subgroup generated by a. Find a presentation for H and show that H ~ 7l.j(3)' (Hint, {1,b,ba} is a Schreier system ofright coset representatives of Hin G.) 246 ApPENDIX I - GROUP THEORY

EXERCISE 6.4 Let G = (a, b I aba = bab, a3 = 1 ). (6.22) Show that the commutator subgroup G' has a presentation (x,y I x2 = y2 = (xy)2).

(Hint, Set x = ba-1 and y = aba- 2 .)

• Triangle groups The but more partieularly quotient groups of the braid group pro• vide a keen area of study and research. Therefore, it is useful to gain some insight into some of the groups that can be shown to be isomorphie to the quotient groups of the braid group. One such example, weIl known to geometers, is the so-called or, equivalently, the polyhedral group.

IDEFINITION 7.1 Suppose l, m and n are integers greater than 1, then the group defined by the presentation

( a, b I al = bm = (abt = 1 ) (7.1) is called the triangle group and denoted by T(l, m, n) .

• Note: It is possible to set one of l, m or n to 1, say l = 1, however, in this case T(l, m, n) reduces to the group 7l/(d), where d = gcd (m, n). As mentioned above, the triangle group is of particular interest to geometers, since this group may be represented as a subgroup (of index 2) of the group generated by 3 reflections along a triangle in the Euclidian plane, ~2, or on the sphere, §2, or on the hyperbolic plane, JH[2, for more exact details see [M*]. (See also Remark 7.1 below.) EXERCISE 7.1 For any permutation {A,J.L, v} of l,m and n prove that

T(l,m,n) = T(A,J.L, v). (7.2) To provide some acquaintance with the triangle group, we shall state 2 theorems concerning triangle groups, and leave the proofs as ambitious exercises or more heipfully refer the reader to [M*].

THEOREM 7.1 The triangle group T(l, m, n), for l, m, n ~ 2, is finite if and only if 111 8 = - + - + - - 1 > O. (7.3) l m n Further, since by (7.3) one of l, m or n must be equal to 2, say l = 2, then the order of a finite triangle group is 4mn 4 - (m - 2)(n - 2)' §7. TRIANGLE GROUPS 247

By some simple arithmetie, we see that there are only four possible types of finite triangle groups:

(1) T(2, 2, n), with n ~ 2, is called the and has order 2n. (2) T(2, 3, 3) is called the tetrahedral group and has order 12. (This group can be shown to be isomorphie to A 4 , the alternating group of 4 letters.) (3) T(2, 3, 4) is called the octahedral group and has order 24. (This group can be shown to be isomorphie to S4.) (4) T(2, 3, 5) is isomorphie to As and has order 60.

THEOREM 7.2 Let T(l, m, n) be the triangle group with 6 < 0, the so called hyperbolic case. Then, an element 9 oE T(l, m, n) is oE finite order if and only if 9 is a conjugate oE a,b or ab. EXERCISE 7.2 Let T(l, m, n) be a triangle group. (1) Show that 6 = 0, see (7.3), if and only if

(l, m, n) = (2,3,6), (2,4,4) or (3,3,3).

(2) Show that T(3, 3, 3) is an infinite group. (Hint, Show that the normal closure of ab-1 in T(3, 3, 3) is an infinite group, or see Remark 7.1.) (3) Similarly, show that T(2, 3, 6) and T(2, 4, 4) are infinite groups.

REMARK 7.1 On ~2 take a triangle 6ABC with angles nil, n/m and n/n at A, Band C respectively. Tiles of this triangle completely cover (Le., without any overlap, gap or opening) ~2 if and only if 6 = 0 in (7.3). So, let (l, m, n) be one ofthe cases in Exercise 7.2(1). Suppose ~2 is completely covered by triangles as above, and then fix a particular 6ABC, whieh we shall call the basic triangle. Next, let 0:, ß and , be 3 reflections of ~2 about the lines containing the edges BC, CA and AB, respectively, of the basie triangle. Let a be the application of , followed by ß, so we may denote this as a = ,.ß. Then, a is a rotation of ~2 at A through an angle of (2n)/l. Similarly, b = 0:., and c = ß.o: are, respectively, the rotations of ~2 at Band C through (2n)/m and (2n)/n. Now, consider the group G of rigid motions of ~2 generated by a, band c. It is known, see for example [M*], that G is isomorphie to T(l, m, n). (To see this, show that al = bm = cn = 1 and abc = 1.) In the particular case of (l, m, n) = (3,3,3), ab-1 represents a parallel re• placement of ~2, and hence, the subgroup generated by ab-1 cannot be finite. Therefore, T(3, 3, 3) is an infinite group. Using similar methods, we can show that T(2, 3, 6) and T(2, 4, 4) are infinite groups. Homotopy Manifolds

• Fundamental concepts of Topology In a nutshell, but without much enlightenment, the field of Topology concerns itself with topological spaces and the continuous mappings between them. Firstly, let us define the notion of a topological space.

DEFINmON 1.1 A topological space is a non-empty set X with the stipulation that Ito each element P (called a point) of X we may assign a family of subsets of X (called neighbourhoods of P) that satisfy the following 4 conditions: (I) A neighbourhood of P always contains the point P. (2) The intersection of 2 neighbourhoods of P is also a neighbourhood of P. (3) If a subset U contains a neighbourhood of P, then U is also a neighbour- hood of P. (4) If U is a neighbourhood of P, then there is a neighbourhood W of P such that U is a neighbourhood of any point of W.

Probably the most familiar example of a topological space is the Euclidian space, jRn, where we think of jRn as the set of n-tuples (Xl,X2, ... ,xn ) with real entries. Since the topological spaces in the body of this book are exclusively Euclidian spaces, we shall restrict ourselves to Euclidian spaces, jRn, and hence tailor this exposition on topological spaces accordingly. For a more general theory, we refer the reader to [Ma *]. In actuality, what exactly is a neighbourhood of P in jRn? To begin with, suppose A = (al,a2, ... ,an) and B = (bl,b2, ... ,bn) are 2 points in jRn. The distance between A and B, denoted by d(A,B}, is defined as

IDEFINmON 1.2 Given a point P in jRn, for any real number 8 > 0, the set

{Q I d(P,Q} < 8} is defined to be a neighbourhood of P, and denoted by U(P} or, equivalently, I. UR" (P). (Note, U(P) contains P.}

The set jRn with the above family of neighbourhoods is a topological space.

248 §l. FUNDAMENTAL CONCEPTS OF TOPOLOGY 249

Let V be a subset of ~n. We may, in a very natural fashion, restrict the neighbourhoods of ~n to the subset V, namely, let

Uv(P) = U(P) n V. (1.2)

Then, elearly, these neighbourhoods form a family of neighbourhoods of P in V. Hence, V becomes a topological space, and is called a subspace of ~n. In fact, most of the topological spaces found in the main body of this book are subspaces of ~n. Let X and Y be topological spaces. Then, a quick way of finding a new topological space is to define their product. Let us denote by X x Y the set of all pairs (P, Q), where PE X and Q E Y. A neighbourhood of (P, Q) is a pair (V, W), where V is a neighbourhood of P in X and W is a neighbourhood of Q in Y. With this set of neighbourhoods X x Y becomes a topological space. If X is a subspace of ~n and Y is a subspace of ~m, then X x Y is a subspace of ~n x ~m, which can be shown to be ~n+m.

EXAMPLE 1.1 Denote by I the interval [0,1] E ~1. Then I x I is a subspace in ~2 commonly known as the unit square, and I x I x I is the unit cube in ~3. Now, suppose X and Y are two topological spaces and I is a mapping from X to Y.

IDEFINITION 1.3 Let P be a point in X and Q = I(P). Then, I is said to be con• tinuous at P if for any neighbourhood Uy(Q) in Y there exists a neighbourhood Ux(P) of P in X such that every point of Ux(P) is mapped into Uy(Q). I The mapping I, itself, is said to be continuous if I is continuous at every point of X.

X -f

Figure 1.1 A way of thinking of a continuous mapping I at P is to consider a point P' that is "elose" to P, then the image I(PI ) should also be "elose" to I(P). If a continuous mapping I from X to Y is both onto and one-to-one, then the inverse mapping of I from Y to X, denoted by 1-1, is defined. In addition, if the mapping 1-1 is continuous then I is called a homeomorphism from X to Y. We say that X is homeomorphic to Y if there exists a homeomorphism from X to Y. On the other hand, if X is homeomorphic to a subspace Z of Y, then we say X can be embedded in Y and I is called an embedding of X into Y. It is often useful to restrict, in the natural way using (1.2), a mapping I from X to Y to a mapping from the subspace V (of X) to Y, and denote it by I Iv . 250 ApPENDIX 11 - TOPOLOGY

EXERCISE 1.1 Show the relation "is homeomorphic to" is an equivalence relation. EXAMPLE 1.2 (1) Let f be a mapping from ~1 to ~1 defined by f(x) = x2, for x E ~1. Then, f is continuous but not onto, since f(x) :I -1 for any x E ~1. (2) Let f be a mapping from ~2 to ~2 defined by f(x, y) = (2x + 3y, x - y), then f is a homeomorphism. (Confirm that f-l(X,y) = (~(x + 3y), ~(x - 2y)).) (3) Again, let f be a mapping from ~2 to ~2 but now defined by f(x, y) = (2x2 - y2, 3xy), then f is continuous and onto, but not one-to-one (Why?).

11 Homotopy Besides homeomorphisms, there are other ways of categorizing mappings from one topological space to another.

IDEFINITION 2.1 Let f and 9 be two continuous mappings from X to Y. We say f is homotopic to g, denoted by f ~ g, if there exists a third continuous mapping h : X x [0, 1J -----+ Y, such that

h(x, 0) = f(x) and h(x, 1) = g(x). (2.1)

ISuch a continuous mapping h is called a homotopy between fand g. If we write h(x, t) = ht(x), then ht(x) is a continuous mapping from X to Y for each t in the range 0 ~ t ~ 1, also ho(x) = f(x) and hl(X) = g(x). We say that {ht (x), 0 ~ t ~ I} is a family of continuous mappings from X to Y that connects f and g. If we fix a point x E X, then the mapping ht(x) : [O,lJ -----+ Y is also continuous with respect to t. EXERCISE 2.1 Show that the relation "is homotopic to" is an equivalence relation.

EXAMPLE 2.1 Let fand 9 be continuous mappings from ~1 to itself, given by

f(x) = 2x and g(x) = 3x. (2.2)

Then, fand 9 are homotopic. In fact, the third mapping h : ~1 x [0, 1J -----+ ~1 is given by h(x, t) = xt + 2x. (2.3)

EXERCISE 2.2 Show that f(x) = x2 and g(x) = 2x - 1 are homotopic. IDEFINITION 2.2 Suppose fand 9 are homotopic mappings from X to Y and h is the homotopy. Further, let Z be a subspace of X. In addition, let us suppose: (1) For each point Z E Z, f(z) = g(z). (2) For each value t with 0 ~ t ~ 1 and z E Z, h(z, t) = f(z) (= g(z)). IThen, fis said to be homotopic to 9 relative to Z, and denoted by f ~ 9 (rei Z).

EXAMPLE 2.2 In Example 2.1, f ~ 9 (rei 0), Le., f is homotopic to 9 relative to the origin. §3. FUNDAMENTAL GROUP 251

IDEFINITION 2.3 Let land 9 be 2 embeddings of X into Y. Then, we say I is isotopic to 9 if there is a third embedding H: X x [0,1] ---+ Y x [0,1] such that, for x E X and 0 ~ t ~ 1,

H(x, t) = (h(x, t), t) with h(x,O) = I(x) and h(x, 1) = g(x). I H is called an isotopy between land g. EXERCISE 2.3 Show "is isotopic to" is an equivalence relation .

• Fundamental group As usual, let X be a topological space. X is said to be arcwise-connected if for any two points P and Q in X, there exists a continuous mapping cp: [0,1] ---+ X such that cp(O) = P and cp(l) = Q. (Intuitively, the points P and Q are joined by means of an arc.) Further , X is said to be contractible if any continuous mapping cp from the unit cirde §l to X is homotopic to a constant mapping from §l to X. By a constant mapping, we mean a continuous mapping that sends every point of §l to the same point in X. A continuous mapping I : [0,1] ---+ X is called a closed path or, equivalently, a loop if 1(0) = 1(1). Namely, I is a loop if its image is a dosed curve. Now, let us choose a point Xo in X and fix this point. Let us look at all the loops I: [0,1] ---+ X such that 1(0) = 1(1) = xo, we shall denote this set ofloops by C. For convenience, we shall say such loops are loops with base point xo. We say a loop I (of X) with base point Xo is equivalent to a loop 9 also with base point Xo if I is homotopic to 9 (rel xo). We shall denote such an equivalence by I rv 9 and leave it as an exercise for the reader to show that this, in fact, is an equivalence relation. Since the above defines an equivalence relation on C, we may divide C into equivalence classes, which we shall denote by C* (= C / rv ). Also, let us denote by [f] the equivalence dass of f in C*. PROPOSITION 3.1 C* is a group with the product defined as follows. If I and 9 are two loops with base point xo, then fog is a loop in X given by

f(2t) for 0 ~ t ~ 1j2 { (3.1) (f 0 g)(t) = g(2t _ 1) for 1/2 ~ t ~ 1 and so we define [I] 0 [g] = [log]. The constant mapping t : [0,1] ---+ X, given by t(t) = xo, is the identity for C*. The inverse for an element [I] E C*, is given by the equivalence dass of the mapping 1*, where 1* is the mapping defined as

f*(t) = 1(1 - t). (3.2)

For the sake of consistency, we will write [/r 1 for [1*]. 252 ApPENDIX II - TOPOLOGY

PROOF Check that

[/Ht] = [tH/] = [I], (3.3) [rH/] = [IHr] = [tl· (3.4) o EXERCISE 3.1 Show that a loop I in X with base point Xo is homotopic to the constant mapping (rel xo) if and only if there exists a continuous mapping H: [0,1] x [0, 1] ~ X such that: (1) H(O, t) = H(s, 0) = H(s, 1) = Xo for 0:::;; s, t :::;; 1. (2) H(l, t) = f(t) for 0:::;; t :::;; 1.

DEFINITION 3.1 The group C* is called the fundamental group of X with base point Xo and is denoted by 1I"1(X,XO).

PROPOSITION 3.2 Let X be an arcwise-connected topological space. Tben, tbe fundamental group 11"1 (X, xo) is a topological invariant for X. Tbat is to say, if X is bomeomorpbic to (an arcwise-connected topological space) Y, tben 11"1 (X, xo) ~ 11"1 (Y, Yo), wbere Yo is tbe image of Xo under tbe bomeomorpbism. Suppose X is an arcwise-connected topologieal space and xo, Xl are two ar• bitrary points on X, then 11"1 (X, Xo) is isomorphie to 11"1 (X, Xl). Therefore, the choiee of base point is not relevant for the group structure of 11"1 (X, Xo). Usually, the extra designation of Xo is dropped and the fundamental group of X is written just as 11"1 (X). In general, given a topological space X it is not an easy matter to determine its fundamental group. More often than not, the group structure of the fundamental group is non-abelian and infinite.

EXAMPLE 3.1 Let X = ~n. Then,

(3.5)

We leave it as an exercise to show that every loop in ~n is homotopic to a con• stant mapping.

EXAMPLE 3.2 Let §n be an-dimensional sphere, with n ~ 1. Then, §n is a sub• space of ~n+1 consisting of all points (Xl. X2, ... ,Xn+1) subject to the condition x~ + ... + X~+ 1 = 1. It can be shown, but not easily, that 11"1 (§1) ~ /Z, and, for n ~ 2, 1I"1(§n) = {1}.

EXAMPLE 3.3 Let X be the surface of revolution, about the y-axis, of the circle in ~2 with centre (2,0) and radius 1. The surface X is called a torus (of genus 1). It is known that 11"1 (X, Xo) is a of rank 2, or, equivalently,

(3.6) §4. MANIFOLDS 253

The generators a and b are represented by 2 loops a and ß on X with a = [al and b = [ß]' see Figure 3.1.

Figure 3.1

• Manifolds A typical, but important, example of topological spaces is a manifold. In this last section, we shall discuss some properties of manifolds.

IDEFINITION 4.1 Let X be a topological space (a subspace of ]Rn). A collection of neighbourhoods of X, U = {Ux }, is called a covering of X if the union of all Ux , as the name suggests, covers X, or, precisely, Uu Ux = X. I X is called compact if for any covering U = {Ux } of X, there is a subset of U that has a finite number of members and is also a covering of X.

EXAMPLE 4.1 The closed interval I = [0,1] C]Rl is compact, but the open interval Y = (0,1) of]Rl is not compact. In fact, let us define a neighbourhood Uy(x) of x, with 0< x < 1, as

~ X 3x ( 3X) if 3x < 1 and ( 1) 'f 1 (4.1) 2' 2 2 2' 1 2 ;;::,

Then,

U = {Uy(~),Uy (~) ,Uy (212 )"" ,Uy (2~)""} (4.2) is an infinite collection of neighbourhoods that covers Y, but it is not possible to find a finite subset of U that will cover Y.

Let ]R+' be a upper half space of ]Rn, Le.,

(4.3)

DEFINITION 4.2 A topological space M is called an-dimensional manifold (or Isimplya n-manifold) iffor each point P of M there is a neighbourhood UMl(P) that is homeomorphic to either ]Rn or ]R+'. The set of points, {P}, in M such that UMl(P) is homeomorphic to ]R+' is I. called the boundary of M and is denoted by aM. 254 ApPENDIX II - TOPOLOGY

EXAMPLE 4.2 (1) The closed interval I = [0,1] C]RI is a 1-manifold. However, the cross in ]R2, shown in Figure 4.1(a), is not a 1-manifold, since any neighbourhood of the crossing point Pisnot homeomorphic to either ]R I or ]R~. (2) A unit circle §I C]R2 is a 1-manifold. However, the figure 8 (in ]R2), shown in Figure 4.1(b), by the same reasoning as in the first part is not a 1-manifold.

+(a) (b) Figure 4.1 Let us denote a compact (arcwise-) connected n-manifold by Mn. If 8Mn = 0, the empty set, then M[Tl is a called a closed mani/old. However, if 8M[Tl "I- 0 then 8 M[Tl is a closed (n - 1) -manifold. EXAMPLE 4.3 Let Mn be a compact arcwise-connected n-manifold. Then, (1) MI is (homeomorphie tO) either a closed interval 1= [0,1] or §I. The closed interval I has a non-empty boundary consisting of the 2 points {O} and {1}. But, §I is a closed 1-manifold. (Note, a single point is a O-manifold.) (2) ~ is usually called a sur/ace. §2 and the torus, considered in Examples 3.2 and 3.3, are closed surfaces. While, the unit disk in ]R2, given by

(4.4) is a compact 2-manifold with a boundary that is (homeomorphic to) §1. We can construct all closed surfaces systematieally. In fact, any closed surface can be obtained from ][)2 by identifying a pair of points on the boundary of ][)2.

EXAMPLE 4.4 (1) Divide the unit circle by 4n (n ~ 2) equidistant points Ao,At, ... ,~n-I. Let us identify two points Pk and Pie on ~kA4k+1 and A 4k+2A4k+3, respectively, that are symmetrie with respect to the Une () = !~ + 2:11" , for k = 0,1, ... , n - 1. Similarly, let us identify two points Qk and Qk on A4k+1A4k+2 and ~k+3A4k+4, respectively, that are symmetrie with respect to the Une () = ~~ + 2:11", for k = 0,1, ... , n - 1. Then, the resultant surface is the orientable closed surface 'll' n of genus n.

(2) Divide the unit circle by 2n (n ~ 1) equidistant points Ao,At, ... ,A2n- I . But in this case, identify two points Pk and Pie on A2kA2k+1 and A2k+1A2k+2, for k = 0,1, ... ,n - 1, that correspond to (1,0 + 2:11") and (1,0 + (2k!I)11"), respectively, with 0 ~ 0 ~ *. Then, the resultant surface is a non-orientable closed surface and denoted by Nn . If n = 1, NI is called the projective plane and is denoted by p2 . • Note: In either case, Ai, for i = 0,1, ... , is identified with the single point Ao. It is known that any closed surface lF is one and only one of the surfaces constructed in Example 4.4. §4. MANIFOLDS 255

As might be expected, the fundamental group of a closed surface can be neatly written down. THEOREM 4.1 In the case of an orientable c10sed surface,

(4.5) while, 11'1 ('Jrn ) bas a presentation:

2n generators: al, a2, ... ,an, bb b2, ... ,bn

1 relation: [ab bl ][a2, b21 ... [an, bnl = 1 wbere [ai, bil = aibia;lb;l for i = 1,2, ... ,n. Tbe generators ab a2, ... ,an and bb b2, ... ,bn, are, respectively, represented by 2n loops ab a2, ... ,an, and ßb ß2, ... ,ßn witb base point Xo, see also Fig• ure 4.2. In tbe case of an non-orientable c10sed surface,

(4.6)

In general, 1I'1(l~n), with n ~ 1, bas a presentation n generators: Xl,X2, .•. ,Xn , 1 relation: x~ x~ ... x~ = 1.

Figure 4.2

EXERCISE 4.1 Prove that 11'1 (f~h) has the following presentation,

(4.7)

N2 is usually called the Klein bottle.

EXERCISE 4.2 Let Pb P2 , ••• Pn be n distinct points in a unit disk 11))2. Then, show that 11'1 (11))2 - {PI, P2 , ••• Pn }) is a Iree group of rank n and specify its generators. A compact surface lF* with boundary is obtained from a closed surface lF by cutting it finitely many times along disjoint simple closed curves on lF. For exam• pIe, a cylinder eisa compact surface with 2 circles as its boundary. C is obtained by cutting the torus 1fl (of genus 1) along the curve a (or ß) in Figure 3.1. • Symplectic group To begin with, let us consider the set of all 2n x 2n matrices whose entries are elements ofthe field 7l. / (P) (= {O, 1, 2, . .. ,p - I} ), where p is a prime number. Further , let us restrict these matrices to the ones with determinant 1. With the addition of this restriction, the considered set becomes a group under the usual matrix multiplication and it is usually called the special and denoted by sl(2n, 7l./(P)) . • Note: For convenience, in 7l./(P) we shall consider the number p - 1 as -1 (since p - 1 == -1 (mod p)).

IDEFINITION 1.1 A 2nx2n matrix, M, in sl(2n, 7l./(P))' whosetransposewedenote by MT, is called a symplectic matrix if MT J M = J, where J=[-±ti-l (1.1) Iand In is the usual n x n identity matrix, since, by the above note, we may replace p-lby-l.

PROPOSITION 1.1 The set of aJl 2n x 2n symplectic matrices forms a subgroup of sl(2n,7l./(p)), called the symplectic group and denoted by sp(2n, 7l./(P)).

PROOF Let A and B be symplectic matrices in sp(2n, 7l./(p)). Then,

(AB)T J(AB) = BT AT JAB = BT JB = J, (1.2) and hence AB is in sp(2n, 7l./(P)). Further, suppose A is in sp(2n, 7l./(p)), then AT JA = J implies

J = (AT)-lJA-i = (A-i)T JA-i. (1.3)

Hence, A-i is in sp(2n, 7l./(p)). Therefore, sp(2n, 7l./(P)) is a subgroup of sl(2n, 7l./(p)). o

EXERCISE 1.1 Show that sp(2, 7l./(p)) = sl(2, 7l./(p)).

256 §l. SYMPLECTIC GROUP 257

The following proposition is well-known, see for example [D*]. PROPOSITION 1.2 For a prime p ~ 2, the centre, C, of sp(2n, d:./(p)) is {hn, -I2n }. Further, if n ~ 2 then sp(2n, d:./(p)) /C is a simple group called the projective symplectic group and denoted by Psp(2n, d:./(p)). If p ~ 3, the order of Psp(2n, d:./(p)) is given by

IPsp(2n, d:./(p))1 = ~pn2(p2 _1)(p4 -1) ... (p2n -1). (1.4)

EXAMPLE 1.1 For p = 3 and n = 2, the order of Psp( 4, d:./(3)) is 25,920. To study sp( 4, d:. / (p) ), we need to define it in a slightly different manner. Firstly, let us define matrices A and B as

0100] A = Ö g1 dB= [ 0001. (1.5) [~1o -1 0~ 1 an 0010 o 0 -1 0 -1 0 0 1 Simple calculations, which we leave as exercises for the reader, show that A belongs to sl(4, d:./(p)), detB = -1 and BT AB = J. PROPOSITION 1.3 Let W be the set of all matrices, C, in sl (4, d:. /(p)) such that CTAC=A. (1.6) Then, W is a subgroup of sp( 4, d:. /(p)) and, further, the inner automorphism c.p: sl(4, d:./(p)) ---> sl(4, d:./(p)) defined, for M in sl(4, d:./(p)), by c.p(M)=B-1MB (1.7) induces an isomorphism {rom W onto sp( 4, d:. /(p) ). PROOF Firstly, for M in sl(4, d:./(p)) such that MT AM = A, it follows from the definition that (1.8) Hence, BTMT(BT)-lJB-1MB = J. (1.9) Therefore, c.p(M) = B-1MB is in sp(4, d:./(p)), and so, c.p maps W into sp( 4, d:. / (p)). Secondly, for an arbitrary N in sp( 4, d:. / (p) ), (BNB-1f A(BNB-1) = (BT)-lNTBT ABNB-1 = (BT)-lNT JNB-1 = (BT)-lJB- 1 = A. (1.10) Hence, c.p-l(N) = BNB-1 is in W. Therefore, by means of c.p, we can identify sp(4, d:./(p)) with W. o 258 ApPENDIX III - SYMPLECTIC GROUP

EXAMPLE 1.2 Let 0'1,0'2,0'3 and 0'4 be Artin generators of the 5-braid group, B 5. Then, the mapping p: B 5 ---+ 8l(4, 'R../(P», defined by

0'2 ---+p 10] , [-\~ 10 01

induces a homomorphism from B5 to 8l(4, 'R../(P», where the empty spaces in the above matriees correspond to the element O. EXERCISE 1.2 (1) Show that the mapping defined above does indeed induce a homomorphism from B 5 to 8l(4, 'R../(p». (2) Show that for i = 1,2,3, and 4 and the matrix A as in (1.5),

p(O'i)T Ap(O'i) = A. (1.11) Hence, show that P(O'i) belongs to W. (3) Show that (1.12)

For the time being, let us assume p = 3. Further, let Ti = P(O'i) for i = 1,2,3, and 4. Then, the following proposition holds, [As] and[W], see also Example 1.2. PROPOSITION 1.4 The group W, and hence 8p(4, 'R../(3», is generated by Tl,T2,T3 and T4 and has the following four relations: (1) TiTi+1Ti = Ti+11iTi+1 for i = 1,2,3. (2) TiTj = TjTi for i,j = 1,2,3,4 and li - jl ~ 2. (3) Tl = h for i = 1,2,3,4. (4) (T1T2T3T4)10 = 14 .

Furthermore, it can be shown that (T1T2T3T4)5 generates the centre of 8p(4, 'R../(3»' Therefore, if N is the normal closure of (T1T2T3T4)5, then W/N is isomorphie to the projective symplectic group P8p(4, 'R../(3» and has order 25,920 (Example 1.1).

EXERCISE 1.3 (1) Show that in B5 (3) (= B 5/(O'r»

0'30'1 (0'20'3) -30'1 (0'20'3)3(0'30'4) -3(0'20'3) -30'1 (0'20'3)3 (0'30'4)3 = (0'10'20'30'4)10. (1.13) (2) Show that also in B 5 (3) (0'10'20'3)4(0'4'10'30'4'10'2"10'30'2"1 )0'1 (0'20'3'10'20'40'3'10'4) = (0'10'20'30'4)10. (1.14) Appendix IV

Proof that ( 0"1-1 0"2)10 - 1 . 3 The proof .. - 3 In B (5) cl f, lS glven . g e ormation in (4)_(5)1~S :heconJugation fo.llowin seriesof (0"-10" 1of cli 2 )~~rams, by 0" !igures10"-1 -1 (1)-(14) . The 1 2 0"1 0"-12 0"1-1 . ( W, ?J; - --:<;8 - ~r ~~ ~ ~ (2) r:?-s(3)

-

(7) (8)

259 260 ApPENDIX IV

(12) (13) (14) Appendix V

Proof of Proposition 5.1 in Chapter 5. As was shown in Figures 4.3(a) and (b), we can eliminate 3 consecutive twists by a ~3 -operation, and by the same reasoning we can replace 2 positive twists by 1 negative twist. Hence, 30 ( -1 -1)6 "( = 0"20"30"40"10"2 0"30"40"10"20"30"1 0"20"30"4 -1 -1)( -1 -1 -1)4 = ( 0"20"30"40"10"2 0"30"40"10"20"1 0"20"30"4 0"2 0"30"40"10"2 0"30"40"10"20"1 0"20"30"4

X (0";-10"30"40"10";-10"30"40"10"20"30"1 10"20"30"4), from which it is easy to see that 30 -1 (-1 -1 -1 -1 )6 0"2"( 0"2 = 0"2 0"30"40"10"2 0"30"40"2 0"10"2 0"30"4 . So, let -1 -1 -1-1 A = 0"2 0"30"40"10"2 0"30"40"2 0"10"2 0"30"4· Then, to show "(30 = 1, it suffices to show that A6 = l. To this end, graphically and making use of Exercise 2.4 in Chapter 5, we show in Figures (1)-(10) that A2 can be deformed into 2 -1 -1 -1 -1 -1-1 A = 0"10"30"2 0"30"40"10"3 0"20"3 0"40"1 0"20"30"1 0"2 0"30"4· Next, using Figure (10), we deform, again with the help of Exercise 2.4, A4 = A2 A2 in the sequence of diagrams given in Figures (11)-(17) into 4 - 1 -1 -1 -1 - 1 -1 -1 - 1 A = 0"10"3 0"40"30"2 0"30"4 0"1 0"2 0"30"2 0"30"40"1 0"20"3 0"4· Finally, using Figures (10) and (17) and as before Exercise 2.4, we deform A6 = A2 A4 into the identity braid, Figures (18)-(24). This now proves Proposition 5.1 in Chapter 5. o

-

(1) (2) (3) (4)

261 262 ApPENDIX V

-- -

(5) (6) (7)

(10) (9) (8) ApPENDIX V 263

(11)

(17) (15) (16) 264 ApPENDIX V

(20) (21) (18) (19)

(23) (24)

(22) Bibliography

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268 INDEX 269 commutator elementary subgroup, 30, 85, 198, deformation, 129, 130 226, 241, 243, 246 move, 4-5, 22, 130 compact, 253 embedding, 249 compactification, 128 equivalence, 219, 220 complete braid (plait), see braid class, 14, 220 complete invariant, 107 of braids, 4, 5, 8, 31, completely solvable, 216, 218 96, 97, 105, 106, 107, 114 configuration space, 214 relation, 219, 220 congruent, 219 Euclidian space, 248 conjugacy problem, 43 exceptional level plane, 133 constructible, 71, 72, 212 exponent sum, 29, 95, 120, 121, 127, continuous mapping, 248, 249 142, 145, 196 contractible, 251 conventional moves, 194 F convex, 98 f-point, 136 core, 206 factor group, 226 coset, faithful representation, 168 left ------,224, field, 230 225,243 finite presentation, 235, 245 right ------,224, fractional statistics, 215 225,242 free group, 233, 239, 245 covering, 253 abelian ------, 233 Coxeter, H.S.M., 79 centre of ------, 233 Coxeter-Todd method, 82 rank of , 233 crossing index, 9, 30 fundamental group, 199, 210, 215, 252 cube, 3, 249 cycle, 222 g cyclic relation, 110 generalized linking number, 111 generators 1> Artin ------, 18 decision problem, 60 set of ------, 234 deformation ----- of n-braid group, 17,26 elementary ------, 129, group, 221 130 abelian ,223, type i, ii ------• 152 239 diagram, 8 centre of ,223, knot ------• 130 225,233 regular ------• ,8 commutative ,223 dihedral group, 247 factor ,226 Dirac's finite ,222 braid, 195 free ,233, problem, 191, 192, 195 239,245 dodecahedron, 79-80 infinite ,222 double point, 7 isomorphic ,227 dual polyhedron, see polyhedron non-abelian ,223 dynamical system, 2, 145 non-commutative ,223 order of ,222 E presentation ,234 elementary quotient ,226, homotopy move, 115 237, 246 270 INDEX group knot simple ------, 224 orientation of ------, 129 triangle , 246 torus , 144, 166, 177 H trefoil ------, 164 hexahedron, 79-80 trivial , 129 homeomorphism, 130, 249 ----- diagram, 130 homogeneous braid, 145 ----- invariant, 169 homomorphism, 227 ----- projeetion, 130 homotopie, 115, 124, 250 homotopieally L ----- equivalent, 116 Laurent polynomial, 189, 230 ---- trivial, 117, 119, 122 length of a braid, 145 homotopy, 250 level plane, 3 ---- braid group, 116, 119 exeeptional ------, 133 ----- invariant, 120,121,123 link(s), 186 ---- of braids, 113 n-eomponent -----• ,130 ----- move, 115 ----- trivial ,132 braided ------• , see I braided ieosahedron, 79-80 equivalenee of -----• ,130 identity braid, 14, 15 homotopic ------,127 index, 225 linking index, 10, 30, 56, 111 inner automorphism, 229, 257 loop,251 interseetion point, 7 invariant, 8 see also braid M invertible knot, 166, 187, 188 M-equivalent, 148 isomorphie, 227 manifold, 214 isomorphism, 227 n-dimensional ------, 253 isotopy, 96, 251 braid group of , 215 ambient , 96, closed ,253 97, 99, 107, 116, 124 mapping, 226 s(trong)- , 97, Markov 105, 106, 107, 114 ----- equivalent, 148 ---- move(s), 147-148 J Markov's theorem, 142, 146, 148 Jones polynomial, 178, 179, 185, 186, Mathematica, 79 187, 188, 189 matrix ----- representation, 230 K ----- ring, 230 Kinoshitar-Terasaka knot, 187 meridian, 134 Klein bottle, 255 ---- disk, 134 knot, 128, 132, 135, 136, 143, 146, 148 Mexiean achiral , 164 ----- braid, 58 amphieheiral , 164, ------algorithm, 67 187, 188, 189 ----- braid group, 59, 71 equivalenee of ----- , 129, ----- plaits, 1, 58, 64 ,65, 71 invertible , 166, Milnor 188 ----- invariant, 108 mirror image of ----- , 164, -----number, 108, 109, 124 165, 188 minimal braid presentation, 143, 166 INDEX 271 mirror image of a knot, 164, 165, 187 presentation modulo (mod), 219 ---- of homotopy braid move group, 119 d------, 75, ----- of n-braid group, 18, 84 19 conventional , 194 ----- of pure n-braid group, elementary , 4-5 50,56 elementary homotopy -- , 115 primitive nth root of 1, 216 quasi-elementary , 22 product, 221 Markov, see Markov ----- of braids, 12, 13 projection N knot ------, 130 n-braid, see braid regular , 7, 8, neighbourhood, 248 15 normal projective plane, 218, 254, 255 ----- closure, 225 braid group of , 200, ----- subgroup, 224 202, 204, 205 projective symplectic group, 94 o pure octahedral group, 247 ----- n-braid, 9, 27, 30, 43, octahedron, 79-80 44, 50, 56, 58, 59 ,65, 71, 109, operation, see move 111, 112, 116, 119, 124, 127, 215 order ----- n-braid group, 59, 63, of an element, 225 64, 67, 122, 124, 215 ----- of a group, 222 ---- (n,n)-tangle, 125, 126 orientat ion preserving, 130 p ~ quasi-braid, 133, 138, 141 permutation, 221 quasi-elementary move, 22 braid ------, 9, , 205 120, 143, 185, 200 quaternions, 231 even ------, 223 quotient group(s), 74, 226, 237, 246 odd , 223 ----- of braid group, 79, 81, plaits, see Mexican 82, 84, 89, 90, 94, 95 Platonic solids, 79-80 type of ------,80, 1{ 81 rank, 233 point of intersection, 7 reflective, 220 polyhedral group, 246 see also triangle regular group ----- diagram, 8 polyhedron (regular), 79-80 ----- polyhedron, see dual------, 81 Platonic solids presentation, 234 ----- projection, 7, 15 finite ------, 235 Reidemeister ----- of alternating group,78 ----- moves, 131 ----- of braid group of a ------Schreier method, 37, projective plane, 202 43, 44, 86, 87, 198, 205, 240-243 ----- of braid group of a relation, 235 sphere, 195 defining , 235 ----- of braid group of a relator, 235 torus, 207 representation, 231 272 INDEX representation, surface Burau ,168 braids in , 199 rewriting process, 242-243 fundamental group of closed , 254- right coset representatives, 240 255 ring, 229-230 orientable closed , 254 commutative ------, 230 non-orientable closed--- , 254 matrix ,230 symmetrie, 220 quaternion ------,231 symmetrie group, 27, 75, 76, 77, 197, ----- of Laurent polynomi- 210, 222, 223, 231, 247 als, 230 symplectic ---- group, 256 S ------cent re of, 257 s-isotopy, see isotopy ------projective, 257 s-point, 136 ----- matrix, 256 same braid, 4 Schreier T ----- system, 33, 44, 77, 86, tangle(s), 114, 124, 127 205,241 homotopic , 115 ----- condition, 241 tensor product, 178 separating simple closed curve, 136, 137 tetrahedral group, 247 strong isotopy, see isotopy tetrahedron, 79-80 sennit topological space, 248 English , 73 topology, 248 French ,73 torus, 134, 218, 252, 255 square , 73 braid group of , 200, sign(r), 108 205, 207, 210, 213 sign of crossing point, 138 ----- word problem for, 213 simple group, 94, 224, 257 braids in , 206 solvability solid , 134 ----- of algebraie equations, standard unknotted , 134 214, 215 ---- knot, 144, 166, 177 , 256 transitive, 220 sphere tree-like disk(s), 137 n-dimensional ------, 252 trefoil knot, 164 braid group of , 194, triangle group, 82, 198, 246- 247 195, 196, 197, 198 trivial braid, 14 braids in ------, 194 ----- word problem ,210 U standard unknotted, 129 ---- braided link, 134 unoriented, 129 ----- unknotted torus, 134 upper half space, 253 standard diagram of a braided link, 139 string, see braid W strong isotopy, see isotopy word, 212, 213, 232 subgroup, 223, 224 reduced , 233 commutator , 226 ----- of a braid, 99,103, normal ,224 104, 105, 107, 108, 109, 120 ---- generated by , 225 ----- problem, 31, 42, 60, subspace, 249 107, 131, 210, 238-240 surface, 254 braid group of----- , 199 Other Mathematics and Its Applications titles of interest:

P.H. SeIlers: Combinatorial Complexes. A Mathematical Theory of Algorithms. 1979, 200 pp. ISBN 90-277-1000-7 P.M. Cohn: Universal Algebra. 1981,432 pp. ISBN 90-277-1213-1 (hb), ISBN 90-277-1254-9 (pb) J. Mockor: Groups ofDivisibility. 1983,192 pp. ISBN 90-277-1539-4 A. Wwarynczyk: Group Representations and Special Functions. 1986, 704 pp. ISBN 90-277-2294-3 (pb), ISBN 90-277-1269-7 (hb) I. Bucur: Selected Topics in Algebra and its Interrelations with Logic, Number Theory and Algebraic Geometry. 1984,416 pp. ISBN 90-277-1671-4 H. Walther: Ten Applications ofGraph Theory. 1985,264 pp. ISBN 90-277-1599-8 L. Beran: Orthomodular Lattices. Algebraic Approach. 1985,416 pp. ISBN 90-277-1715-X A. Pazman: Foundations of Optimum Experimental Design. 1986, 248 pp. ISBN 90-277-1865-2 K. Wagner and G. Wechsung: Computational Complexity. 1986,552 pp. ISBN 90-277-2146-7 A.N. Philippou, G.E. Bergum and A.F. Horodam (eds.): Fibonacci Numbers and Their Applications. 1986,328 pp. ISBN 90-277-2234-X C. N astasescu and F. van Oystaeyen: Dimensions ofRing Theory. 1987, 372 pp. ISBN 90-277-2461-X Shang-Ching Chou: Mechanical Geometry Theorem Proving. 1987, 376 pp. ISBN 90-277 -2650-7 D. Przeworska-Rolewicz: Algebraic Analysis. 1988, 640 pp. ISBN 90-277-2443-1 C.T.J. Dodson: Categories, Bundles and Spacetime Topology. 1988,264 pp. ISBN 90-277-2771-6 Y.D. Goppa: Geometry and Codes. 1988,168 pp. ISBN 90-277-2776-7 A.A. Markov and N.M. Nagorny: The Theory ofAlgorithms. 1988,396 pp. ISBN 90-277-2773-2 E. Kratzei: Lattke Points. 1989,322 pp. ISBN 90-277-2733-3 A.M.W. Glass and W.Ch. Holland (eds.): Lattice-Ordered Groups. Advances and Tech• niques. 1989,400 pp. ISBN 0-7923-0116-1 N.E. Hurt: Phase Retrieval and Zero Crossings: Mathematical Methods in Image Recon• struction. 1989,320 pp. ISBN 0-7923-0210-9 Du Dingzhu and Hu Guoding (eds.): Combinatorics, Computing and Complexity. 1989, 248 pp. ISBN 0-7923-0308-3 Other Mathematics and lts Applications titles of interest:

A. Ya. Helemskii: The H omology ofBanach and Topological Algebras. 1989, 356 pp. ISBN 0-7923-0217-6 J. Martinez (ed.): Ordered Algebraic Structures. 1989,304 pp. ISBN 0-7923-0489-6 V.I. Varshavsky: Self Timed Control of Concurrent Processes. The Design of Aperiodic Logical Circuits in Computers and Discrete Systems. 1989,428 pp. ISBN 0-7923-0525-6 E. Goles and S. Martinez: Neural and Automata Networks. Dynamical Behavior and Applications. 1990,264 pp. ISBN 0-7923-0632-5 A. Crumeyrolle: Orthogonal and Symplectic Clifford Algebras. Spinor Structures. 1990, 364 pp. ISBN 0-7923-0541-8 S. Albeverio, Ph. Blanchard and D. Testard (eds.): Stoehastics, Algebra and Analysis in Classical and Quantum Dynamics. 1990, 264 pp. ISBN 0-7923-0637-6 G. Karpilovsky: Symmetric and G-Algebras. With Applications to Group Representations. 1990,384 pp. ISBN 0-7923-0761-5 J. Bosak: Deeomposition ofGraphs. 1990,268 pp. ISBN 0-7923-0747-X J. Adamek and V. Trnkova: Automata and Algebras in Categories. 1990, 488 pp. ISBN 0-7923-0010-6 A.B. Venkov: Speetral Theory ofAutomorphie Funetions and lts Applications. 1991,280 pp. ISBN 0-7923-0487-X M.A. Tsfasman and S.G. Vladuts: Algebraie Geometrie Codes. 1991,668 pp. ISBN 0-7923-0727-5 H.J. Voss: Cycles and Bridges in Graphs. 1991,288 pp. ISBN 0-7923-0899-9 V.K. Kharchenko: Automorphisms andDerivations ofAssociative Rings. 1991,386 pp. ISBN 0-7923-1382-8 A.Yu. Olshanskii: Geometry ofDefining Relations in Groups. 1991,513 pp. ISBN 0-7923-1394-1 F. Brackx and D. Constales: Computer Algebra with L/SP and REDUCE. An lntroduction to Computer-Aided Pure Mathematics. 1992, 286 pp. ISBN 0-7923-1441-7 N.M. Korobov: Exponential Sums and their Applications. 1992,210 pp. ISBN 0-7923-1647-9 D.G. Skordev: Computability in Combinatory Spaces. An Algebraic Generalization of Abstract First Order Computability. 1992, 320 pp. ISBN 0-7923-1576-6 E. Goles and S. Martinez: Statistical Physics, Automata Networks and Dynamieal Systems. 1992,208 pp. ISBN 0-7923-1595-2 M.A. Frumkin: Systolie Computations. 1992,320 pp. ISBN 0-7923-1708-4 J. Alajbegovic and 1. Mockor: Approximation Theorems in Commutative Algebra. 1992, 330 pp. ISBN 0-7923-1948-6 Other Mathematics and Its Applications tides of interest:

LA. Faradzev, A.A. Ivanov, M.M. Klin and AJ. Woldar: Investigations in Algebraic Theory ofCombinatorialObjects. 1993,516 pp. ISBN 0-7923-1927-3 LE. Shparlinski: Computational and Algorithmic Problems in Finite Fields. 1992, 266 pp. ISBN 0-7923-2057-3 P. Feinsilver and R. Schott: Algebraic Structures and Operator Calculus. Vol. I. Represent• ations and Probability Theory. 1993,224 pp. ISBN 0-7923-2116-2 A.G. Pinus: Boolean Constructions in Universal Algebras. 1993,350 pp. ISBN 0-7923-2117-0 V.V. Alexandrov and N.D. Gorsky: Image Representation and Processing. A Recursive Approach. 1993,200 pp. ISBN 0-7923-2136-7 L.A. Bokut' and G.P. Kukin: Algorithmic and Combinatorial Algebra. 1994,384 pp. ISBN 0-7923-2313-0 Y. Bahturin: Basic Structures of Modern Algebra. 1993,419 pp. ISBN 0-7923-2459-5 R. Krichevsky: Universal Compression and Retrieval. 1994,219 pp. ISBN 0-7923-2672-5 A. Elduque and H.C. Myung: Mutations ofAlternative Algebras. 1994, 226 pp. ISBN 0-7923-2735-7 E. Goles and S. Martnez (eds.): Cellular Automata. Dynamical Systems and Neural Net• works. 1994,189 pp. ISBN 0-7923-2772-1 A.G. Kusraev and S.S. Kutateladze: Nonstandard Methods ofAnalysis. 1994,444 pp. ISBN 0-7923-2892-2 P. Feinsilver and R. Schott: Algebraic Structures and Operator Calculus. Vol. II. Special Functions and Computer Science. 1994,148 pp. ISBN 0-7923-2921-X V.M. Kopytov and N. Ya. Medvedev: The Theory of Lattice-Ordered Groups. 1994,400 pp. ISBN 0-7923-3169-9 H. Inassaridze: Algebraic K-Theory. 1995,438 pp. ISBN 0-7923-3185-0 C. Mortensen: Inconsistent Mathematics. 1995,155 pp. ISBN 0-7923-3186-9 R. Ablamowicz and P. Lounesto (eds.): Clifford Algebras and Spinor Structures. A Special Volume Dedicated to the Memory of Albert Crumeyrolle (1919-1992).1995,421 pp. ISBN 0-7923-3366-7 W. Bosma and A. van der Poorten (eds.), Computational Algebra and Number Theory. 1995,336 pp. ISBN 0-7923-3501-5 A.L. Rosenberg: Noncommutative Algebraic Geometry and Representations of Quantized Algebras. 1995,316 pp. ISBN 0-7923-3575-9 L. Yanpei: Embeddability in Graphs. 1995,400 pp. ISBN 0-7923-3648-8 B.S. Stechkin and V.L Baranov: Extremal Combinatorial Problems and Their Applications. 1995,205 pp. ISBN 0-7923-3631-3 Other Mathematics and Its Applications titles of interest:

Y. Fong, H.E. Bell, w.-F. Ke, G. Mason and G. Pilz (eds.): Near-Rings and Near-Fields. 1995,278 pp. ISBN 0-7923-3635-6 A. Facchini and C. Menini (eds.): Abelian Groups and Modules. (proceedings of the Padova Conference, Padova, Italy, June 23-July 1, 1994). 1995,537 pp. ISBN 0-7923-3756-5 D. Dikranjan and W. Tholen: Categorical Structure ofClosure Operators. With Applica• tions to Topology, Algebra and Discrete Mathematics. 1995, 376 pp. ISBN 0-7923-3772-7 A.D. Korshunov (ed.): Discrete Analysis and Operations Research. 1996,351 pp. ISBN 0-7923-3866-9 P. Feinsilver and R. Schott: Algebraic Structures and Operator Calculus. Vol. III: Repres• entations ofLie Groups. 1996,238 pp. ISBN 0-7923-3834-0 M. Gasca and C.A. Micchelli (eds.): Total Positivity and Its Applications. 1996,528 pp. ISBN 0-7923-3924-X W.D. Wallis (ed.): Computational and Constructive Design Theory. 1996,368 pp. ISBN 0-7923-4015-9 F. Cacace and G. Lamperti: Advanced Relational Programming. 1996, 410 pp. ISBN 0-7923-4081-7 N.M. Martin and S. Pollard: Closure Spaces and Logic. 1996,248 pp. ISBN 0-7923-4110-4 A.D. Korshunov (ed.): Operations Research and Discrete Analysis. 1997,340 pp. ISBN 0-7923-4334-4 W.D. Wallis: One-Factorizations. 1997,256 pp. ISBN 0-7923-4323-9 G. Weaver: Henkin-Keisler Models. 1997,266 pp. ISBN 0-7923-4366-2 V.N. Kolokoltsov and V.P. Maslov: I dempotent Analysis and Its Applications. 1997, 318 pp. ISBN 0-7923-4509-6 J.P. Ward: Quaternions and Cayley Numbers. Algebra and Applications. 1997,250 pp. ISBN 0-7923-4513-4 E.S. Ljapin and A.E. Evseev: The Theory ofPartial Algebraic Operations. 1997,245 pp. ISBN 0-7923-4609-2 S. Ayupov, A. Rakhimov and S. Usmanov: Jordan, Real and Lie Structures in Operator Algebras. 1997,235 pp. ISBN 0-7923-4684-X A. Khrennikov: Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models. 1997, 389 pp. ISBN 0-7923-4800-1 G. Saad and MJ. Thomsen (eds.): Nearrings, Nearfields and K-Loops. (Proceedings ofthe Conference on Nearrings and Nearfields, Hamburg, Germany. July 30-August 6, 1995). 1997,458 pp. ISBN 0-7923-4799-4 Other Mathematics and Its Applications titles of interest:

L.A. Lambe and D.E. Radford: Introduction to the Quantum Yang-Baxter Equation and Quantum Groups: An Algebraic Approach. 1997, 314 pp. ISBN 0-7923-4721-8 H. Inassaridze: Non-Abelian and Its Applications. 1997,271 pp. ISBN 0-7923-4718-8 B.P. Kornrakov, I.S. Krasil'shchik, G.L. Litvinov and A.B. Sossinsky (eds.): Lie Groups and Lie Algebras. Their Representations, Generalisations and Applications. 1998,358 pp. ISBN 0-7923-4916-4 A.K. Prykarpatsky and LV. Mykytiuk (eds.): Algebraic Integrability of Nonlinear Dynam• ical Systems on Manifolds. Classical and Quantum Aspects. 1998, 554 pp. ISBN 0-7923-5090-1 A.A. Tuganbaev: Semidistributive Modules and Rings. 1998,362 pp. ISBN 0-7923-5209-2 M.V. Kondratieva, A.B. Levin, A.V. Mikhalev and E.V. Pankratiev: Differential and Dif• ference Dimension Polynomials. 1999,436 pp. ISBN 0-7923-5484-2 K. Yang: Meromorphic Functions and Projective Curves. 1999,202 pp. ISBN 0-7923-5505-9 V. Kolmanovskii and A. Myshkis: Introduction to the Theory andApplications ofFunctional Differential Equations. 1999,664 pp. ISBN 0-7923-5504-0 K. Murasugi and B.I. Kurpita: A Study ofBraids. 1999,282 pp. ISBN 0-7923-5767-1 J.S. Golan: Power Algebras over Semirings. With Applications in Mathematics and Com• puter Science. 1999,214 pp. ISBN 0-7923-5834-1