On the rank of quadratic equations in free groups

On the rank of quadratic equations in free groups

Journal of Pure and Applied Algebra 21 60 (1989) 21-31 North-Holland ON THE RANK OF QUADRATIC FREE GROUPS Leo P. COMERFORD, EQUATIONS IN Jr. ...

713KB Sizes 1 Downloads 17 Views


of Pure and Applied



60 (1989) 21-31






Department of Mathematics, Eastern Illinois University, Charleston, IL 61920, U.S.A.



Department of Mathematics, Mount Saint Vincent University, Halifax, Nova Scotia, Canada B3M 2J6

Communicated Received

by K.W.

25 July



In Memory Let Fd and H be free groups stants).

freely generated

An equation over H is an expression

of Roger

by sets X= {xt, . . . ,xd} (variables)

W= 1 where

W is a reduced


and A (con-

word on Xi1


” ;

this equation is called quadratic if for each i, 15 isd, the total number of occurrences of xi and XT’ in W is zero or two. A solution to an equation W= 1 over His an endomorphism @ of Fd*H whose restriction

to H is the identity

W= 1 is the rank of the free group

rank of an equation A formula earlier


W= 1 is the maximum

for ranks of quadratic formulas

on H and such that

for quadratic

I+‘@= 1. The rank of a solution

[email protected], where rc is the projection of the ranks

@ to

of Fd*H onto Fd, and the

of its solutions.


over free groups



is given, and is shown to include


1. Introduction In [4] Roger Lyndon began the investigation of equations over free groups. In that paper the inner rank of an equation is the central concept in terms of which he presents his results. Lyndon’s choice of the term ‘rank’ intentionally suggests an analogy with the rank of matrix equations. He develops this analogy further in the survey paper [S]. Continuing in the spirit of [4] and [5], we shall emphasize the analogy with linear algebra while developing a rank formula for quadratic equations with constants. Our results generalize work of Lyndon [4], Zieschang [lo], and Piollet [7,8]. In [4] Lyndon demonstrated that the inner rank (or, more simply, the rank) of a quadratic equation without constants is effectively calculable. Later Zieschang * Research support from gratefully acknowledged.


the Natural



0 1989, Elsevier Science Publishers



B.V. (North-Holland)


of Canada



L.P. Comerford, CC. Edmunds

[lo] gave a formula for the rank of Whitehead-minimal quadratic equations without constants and implied a formula for all quadratic equations without constants. In [7] Piollet developed a rank formula for quadratic equations without constants based on the co-initial graph. Recently Razborov [9] has given a rank formula for general systems of equations without constants. After deriving our rank formula, Theorem 5.2, we shall show that Piollet’s when reinterpreted without reference to the Zieschang’s formula for equations without illustrating that the additional complexity modate words with constants.

formula becomes Zieschang’s formula graph and that our formula reduces to constants. Finally we provide examples of our formula is necessary to accom-

2. Preliminaries Let Fd and H be free groups freely generated (or based) by the sets X= {xi, x2, . . . , xd} (dll) and A={a,,az,...}. H enceforth Fd and H will be identified with their canonical images in the free product F,*H. The elements of X ” = {x1,x;‘, . . . ,xd, xi1 } are call e d variables, the elements of H are called constants, and the elements X ” U A ’ I are called letters. The cardinality of a set or sequence S is denoted /S / ; thus the length of a word W in F,*H is denoted 1WI. The empty word is represented by 1. The set gen( W) = {xEX: x or x-’ occurs in W} is the set of generators of W, where W is a word or a sequence of words. Words, or sequences of words, W and U are disjoint if gen( W) fl gen(U) = 0. A product W, W2 ... W, is disjoint if the w’s are pairwise disjoint. An H-endomorphisrn (H-automorphism) is an endomorphism (automorphism) of F,*H fixing H elementwise: these will simply be called H-maps. When a specific H-map is defined, we adopt the convention that the mapping fixes all variables in X not explicitly mentioned in the definition. The projection 7~is the homomorphism from Fd* H into itself fixing Fd elementwise and sending each element of H to 1. If x is a variable and y is a constant letter or a variable with x and y disjoint, the H-automorphism Q : x H xy -’ is called an elementary regular H-map and the Hendomorphism o :x c 1 is called an elementary singular H-map. To simplify the statement of subsequent results, we adopt the otherwise inconsequential convention of including the identity map on F,*H, written 1, as both an elementary regular and an elementary singular H-map. The elementary regular H-maps will play the role of elementary matrices in our analogy. We call a product of elementary regular maps a regular H-map and a product of elementary singular maps a singular Hmap. Given a reduced word W and an elementary H-map Q, defined as above, we say Q is W-attached if W contains a subword (xy)’ ‘. The identity map, I, is defined to be W-attached as well. The elementary singular map o : x H 1 is W-attached if x E gen( W). A W-attached H-map is an H-map ,u =pi ,LI~1.. Pi, where the ,u;‘s are elementary regular or singular H-maps, ,u, is W-attached, and for each i > 1, pi is . ..pi_.-attached. WPl


On the rank of quadratic equations

A word

WE Fd*H is quadratic if each variable

of W occurs exactly twice, as two

Xi’s, as two Xi-l’s, or as an Xi and an Xi-‘. Note that this the number of occurrences of constants. It is important quadratic and p is a W-attached H-map, then Wu is also word WEF,*H contains at least one constant, then W is stants; otherwise W is a word without constants. An equation is an expression


1, *.*,

places no restriction on to observe that if W is quadratic. If a reduced called a word with con-

xd; al,az, . ..) = 1


WE F,*H. Equation (1.1) is an equation with constants, an equation without constants, or a quadratic equation as W is, respectively, a word with con-


stants, a word without constants, or a quadratic word. Equations without constants or with constants could be thought of as ‘homogeneous’ or ‘inhomogeneous’. A solution to equation (1.1) is an H-map, @, sending W to 1. If C$is a solution to W= 1 we call @ a W-solution. If 0 is also a singular map, we call @ a singular W-solution. A word W is consistent if the corresponding equation W= 1 has a solution. Note that by [l, Theorem 2.71, consistency of a quadratic word is effectively decidable. If @ is a W-solution and is W-attached, then we call I$ a W-attached solution. In developing his theory of rank and nullity for equations without constants, Lyndon [4] applies the linear cancellation method of Nielsen (see [6, p. 4-131). His main tool in [4] is a result which decomposes the solutions of an equation without constants into more manageable parts. In our terminology his result becomes the following: 61. If WEF,

and C$is a W-solution, then there exist regular Helementary singular H-maps aIru2, . . . . a,, and an H-map w maps el,e2, . . ..e., such that @Ial~2(32.,. Q, O, is a W-attached solution and C$= el a1 e2 ~7~... Q, a, w. [4, Proposition

Piollet quadratic

[8] refines


word without


decomposition We translate

in the


this result

case that

W is a

as follows:

11. If W is a quadratic

word in Fd, a a W-attached elementary singular H-map, and Q a Wa-attached regular H-map, then there exists a Wattached regular H-map Q’ such that [email protected][email protected]‘a. [S, Proposition

Piollet applies this repeatedly to gather the oi’s of Lyndon’s decomposition together on the right. In [2] the authors show that the proofs of Lyndon and Piollet can be modified and extended to give the following result (see [2, Theorems 2.1 and 3.11): Theorem 2.1. If WE F,*H and @is a W-solution, then there exist a regular H-map Q, a singular H-map a, and an H-map y such that QU is a W-attached solution and @[email protected] 0





3. Rank and nullity If @ is an H-map, subgroup


the rank of $I, denoted rank(@), is the rank of F&n, the (free) the rank of the by {x1 @c, . . . , [email protected]}. If W is consistent,

of Fd generated W= 1, denoted rank(W)


is defined


= max{ rank(@): 0 is a W-solution}.

This maximum clearly exists since the rank of any solution is bounded above by d. In the case H= ( 11, all equations are without constants, z is the identity mapping, and our definition of rank(W) coincides with Lyndon’s concept of ‘inner rank’. In [4], Lyndon proved that the rank of the equation x:x$x;* = 1 is 1. It follows that if A, B, CEF, and A*B*= C2, then A, B, and C are all powers of a common element x E Fd. Thus A = xP, B = x4, C = x’ with p + q = r. The solutions of x:xix;* = 1 can then be characterized as exactly those mappings @ :x1 H xp, x2 ++x q, x3 c x’ where x is a variable which can be replaced by any element of Fd. Thus an inner rank of 1 corresponds to one ‘degree of freedom’ in the above sense. Our definition of rank seems to be the most natural extension of Lyndon’s inner rank for equations with constants. We use a different term only to remind the reader of the distinction between rank and inner rank. Lemma 3.1.

If W is a consistent quadratic word, then rank(W)

= max{rank([email protected]):

,QG is a W-attached solution}.

Proof. Let M,=max{rank(F,eorr): eo is a W-attached solution} and note that rank(W) and M, exist since W is consistent. If @ is a W-solution, then Theorem 2.1 provides the existence of a W-attached solution @a and an H-map w such that thus, rank(@) = rank([email protected]) = rank(Fdeocylr) = @[email protected]~. Note that WII=ZW~; rank([email protected])Irank([email protected]). It follows that rank(W)I On the other 0 hand, each W-attached solution ~a is a W-solution; thus A4,s rank( W). elementary singular HIf rJ=g1(T* ... on is a product of distinct non-identity maps, the nullity of CJ, denoted nullity(a), is n. Clearly rank(x,(Tn,~~(Tn, ... . x,,on) = d - n; thus rank(o) + nullity(o) = d. If W is a consistent word, the nullity of W is defined by nullity(W)

Note that since

= min{nullity(a):

W is consistent,

eo is a W-attached solution, Q regular, cr singular}.


exists by Theorem


Lemma 3.2. If WE F,*H and @a is a W-attached solution (Q regular, o singular), then rank(F,Qa7c) + nullity(a) = d. Proof.

Note first that (TIC= X(T on F,*H.

Since Q is a W-attached



On the rank of quadratic equations


Q =Q~Q~***Q,, where each ei is elementary regular. It follows, for each i, that Q;Z = 7rei and that QiZ induces an elementary Nielsen automorphism of Fd. Furthermore, QX =Q, 7ce2n.+.e,n induces an automorphism of Fd. Selecting (x,(@n)-‘,[email protected]))‘, . . . , xd (Q;n)-’ > as a basis for Fd , we see that rank([email protected])

= rank([email protected]) = rank(x,(e~)~‘ena,x2(@7C)-‘@~a, =rank(x,a,x20,...,

We now come to the analog Proposition


x, 0) = d - nullity(o).

of a very familiar

If WE F,*H, then rank(W)

. . ..~~(~n)-‘~rca)


+ nullity(W)

from linear

0 algebra.

= d.

Proof. Select eo to be a W-attached solution with nullity(o)=nullity( W). Since rank(W) L rank([email protected]), it follows, by Lemma 3.2, that rank(W) L d - nullity(o) = d-nullity(W). On the other hand, by Lemma 3.1, we can choose @a to be a Wattached solution with rank(W) = rank([email protected]). Since nullity(W) I nullity(o), it follows, again by Lemma 3.2, that rank(W) = rank([email protected]) = d - nullity(a) I d - nullity(W). 0

4. The factoring


It is well known (see [6, Proposition 7.6, p. 601 and [l, $31) that there is an attached H-map which puts any quadratic word W, without or with constants, into ‘canonical’ form, as a product of commutators or squares followed by a product of constants all but the first of which are conjugated by variables. Although we shall not use this result, it is useful to note that the canonical form provides a factorization of W into disjoint subwords. Since the rank of an equation is invariant under H-automorphisms, it will prove useful to find disjoint factorizations of W to compute its rank. This is not unlike analyzing a linear transformation by factoring its characteristic polynomial and decomposing its domain into the corresponding invariant subspaces. If x and y are disjoint variables and E and q are elements of { -1, +l} for which E = v implies E = - 1, the word xCyVxy is called a genus word. If z is a variable and U ( # 1) is a consistent, quadratic word disjoint from Z, the word z-l Uz is called an index word. If W is a consistent, quadratic word, there is clearly a maximal, non-negative integer g such that W= G W, where G is a product of g genus words and W, is consistent. Furthermore, there is a maximal, non-negative integer k with W, = U,K where K is a product of k index words and U, is consistent. We call g the formal genus of W, denoted g,, and k the formal index of W, denoted kw. Thus a consistent, quadratic word W can be written uniquely as a disjoint product




C.C. Edmunds

(4.1) where the Gi’s and Kj’S are genus and index words, respectively, and U. is a (possibly empty) consistent, quadratic word. We call (G,, ..., G,) the genus se-

quence of Wand, if Kj = ~~7’qZj, we call (CJO,. . . , U,) the index sequence of W. The singularity of W is defined as singularity(W)

= gw+ Igen(Uo, . . . . Uk)l.

Lemma 4.1 (Factoring lemma). If W is a consistent, quadratic word and a is a singular W-solution, then there exists a regular W-attached H-map (Y such that (i) aa is a W-solution, (ii) every term of the genus sequence of Wcr contains a variable which o sends to 1, and (iii) CTsends every variable occurring in the index sequence of Wa to 1. Proof. We induct on the length of W. If 1W 1=O, then W= 1. Letting a= z (the identity map on Fd* H), the lemma follows immediately. Next suppose 1WI > 0. If o sends every variable of W to 1, we set (x = I and the proof is complete. Henceforth we may assume that there is an XE gen( W) with XCJ=x. Replacing x by x-’ if necessary, W can be written as W= Lx-‘MxR and we say that x spans M

in W. Case 1. If the subword

spanned by x in W is not quadratic, then there exists a variable y and an EE (-1, +l} such that either (I) W=AxPIByCxDy”E or (II) W= Ay”Bx-‘CyDxE, where A, B, C, D, and E are subwords of W. In either case, note that since xo =x, the X-I and x in W must cancel against each other when o is applied. Since y occurs only once between x-l and x, it follows that ya = 1. Subcase I. If W is of form (I), let (Yebe the W-attached H-automorphism defined by the composition (x - xA)(y ++D-‘A-‘y)(x - BDPIAP1x)(y Y yADB_‘C-l) when E = 1 and (x c xA)(y +-+ yAD)(x H C-‘D-‘A-‘x)(y Y B-‘CPIDPIAP1y) when E = -1. Thus Wao=x-lyxy’AD(CB)PEE. Since Wa= 1 and X(T=x, it follows that

(BC)a = 1 and (ADE)o = 1; therefore (AD(CB)-‘E)a = 1. Subcase II. If W is of form (II), let a0 be defined by (y H A-‘y)(x H CAP1x)O, C* yAC-‘D-‘)(x++ xAC-‘D-‘B) when E = 1 and (y c yA)(x H DP’AP1x)(y c C-‘D-‘A-‘y)(x - xADCB) when E = - 1. Here Wao = y’xxlyxA(DC)-‘BE with (CD)a= 1 and (ABE)o= 1. It follows that (A(DC)-‘BE)o= 1. In either subcase, we have a W-attached H-automorphism cxo with [email protected], the disjoint product of a genus word G and a consistent word @‘such that WC = 1. Since W is shorter than W, the induction hypothesis implies the existence of a regular H-map 12 such that (i) &a is a W-solution, (ii) every term of the genus sequence, (Cl, . . . , G,), of WC? contains a variable which 0 sends to 1, and (iii) cr sends every variable occurring in the index sequence, (U,, . . . , U,), of W& to 1. We may assume that 6 fixes any variable not in gen(W). Letting a=cxoB, it follows


On the rank of quadratic equations

that (x is a W-attached


with Wa = G(@b) and, thus,

Waa = 1. It

is clear that the genus and index sequences of Wa are (G, Gi, . . . , Gg) and (UO, . . . , U,). Since the genus word G (= G(x, y)) contains a variable, y, which CJsends to 1, the conclusions of the lemma are easily seen to hold. Case 2. Suppose now that each variable x E gen( W) with XB =x spans a quadratic subword of W. Thus W can be written as W=Lx-‘MxR where M is quadratic. Further suppose, without loss of generality, that the variable x is chosen so that the x-l and x are ‘innermost’ in the sense that uo = 1 for all u E gen(M). Defining a0 by (Y,,:x c xR_‘, it follows that o. is a W-attached H-automorphism with Wao= @‘xPIMx where I&‘= LR. Since Wo= 1 and Ma= 1, we have (LR)a = 1. Since [email protected] is shorter than W, the induction hypothesis implies the existence of a regular H-map 6 such that (i) &a is a W-solution, (ii) every term of the genus sequence, (G,, . . . . G,), of [email protected]& contains a variable which o sends to 1, and (iii) o sends every variable occurring in the index sequence, ( Uo, . . . , U,), of I&‘&to 1. As before, we may assume that B fixes any variable not in gen( [email protected]). Setting (Y= a,&, it follows that a is a W-attached H-automorphism with Wa = (@6)x-‘Mx. Thus aa is a W-solution, the genus and index sequences of Wa are (G,, . . . , G,) and (U,, . . . , U,,M), and, since uo= 1 for every u~gen(M), the conclusions of the lemma follow. 0 Lemma 4.2.

If W is a consistent, quadratic word and c~is a singular W-solution,

then there exists a regular W-attached H-map

a such that singularity(


nullity(a). Proof.


Lemma uo= sequence, (G,, . . . , G,), of U,), of Wa. Clearly then

ao is a W-solution,

4.1, we obtain a W-attached H-automorphism a such that 1 for at least one variable, u, in each G; from the genus Wa, and UCJ= 1 for every u in the index sequence, (U,, . . . , q nullity(o) 2 g + 1gen(U,, . . . , U,) I= singularity( Wa).

5. Rank and nullity formulas We are now ready to prove the nullity


Theorem 5.1 (Nullity Formula). Zf W is a consistent, quadratic word, then nullity( W) = min{ singularity( Wa): a is a W-attached H-automorphism}. Combining Theorem

this with Proposition


is immediate.

Zf Wis a consistent, quadratic word, then rank(W) 0 Wa): a is a W-attached H-automorphism}.

5.2 (Rank Formula).

d - min{ singularity( Proof

3.3, our main

of Theorem



mw= min{singularity(

Wa): a is a






Comerford, C.C. Edmunds

automorphism}. First note that m, exists, since the identity mapping z is a Wattached H-automorphism. The remainder of the proof will be divided into two parts: ( 5 ): nullity(W) 5 m w and (2 ): nullity(W) 2 m,. (5) Select a W-attached H-automorphism (x so that m w= singularity( Wa), and factor Wcl as in Lemma 4.1:


($, ~~)c/~(iillZ;lO;~j) where Gi=x~(‘)_$(‘)xiYi-

Since each Uj is a consistent word, Theorem 2.1 implies the existence of regular Hmaps ej and singular H-maps aj such that each Qjaj is a Uj-attached solution. Let e=eo...ek and a=rl...r,oo ... ok where the ri’s are the singular H-maps defined by 5i : Xi H 1 if c(i) = 1 and pi : yi H 1 otherwise. It follows that ,~o is a Wa-attached solution. Thus, I nullity(a)


= g+ i


I g+ f

j=O =g+


Igen( uj) 1


. . . . uk) 1= singularity(

Wa) = mw.

(2) Let ~a be a W-attached solution such that nullity(W) = nullity(a). Note then that (T is a We-attached solution. Applying Lemma 4.2, there exists a regular WQattached H-automorphism (Ysuch that singularity( Wea) I nullity(a). Since @a is a W-attached H-automorphism, we have = nullity(a)


6. The formulas

of Zieschang

2 singularity(

Wea) 2 mw.


and Piollet

A word in FpH is said to be H-minimal if it is of minimal length among its Hautomorphic images. Zieschang [lo] gives rank formulas for quadratic words, without constants, in canonical form. If Cg represents a product of g disjoint commutators of variables and S, represents a product of n disjoint squares of variables, Zieschang gives the formulas rank(C,) =g and rank(S,) = [n/2], where [i] denotes the greatest integer less than or equal to i. Since rank is invariant under automorphisms and every H-minimal quadratic word without constants has the same length as its canonical form, we can write these two formulas in one. If W is an Hminimal quadratic word without constants, then rank(W) = [I gen( W) //2]. Zieschang also implies a formula, without recording it explicitly, for arbitrary quadratic words without constants. If x is an H-automorphism which brings W into canonical form, then Igen( W)l - lgen( Wx)I is the number of variables lost in passing to canonical, and hence H-minimal, form. It follows that for a quadratic word W without constants, rank(W) If we suppose


[Igen(Wx)lA + (IgdW)I

that we are in the context

- Igen(Wx

of Fd, this formula


On the rank of quadratic equations


rank(W) = [Igen(Wx)l/2l+_(lgen(W)( - /gen(Wx)l)+(d- lgen(W)l)


Igen(Wx)l+ 1





We also record that, by Proposition following nullity formula: nullity(W) W quadratic



3.3, Zieschang’s





Isn(Wx)I + 1




[ For




[7] gives the rank


= [ “‘W2-l],

where p(W)= 1Wl/2+ I& and lzwl represents the number of connected components of &, the co-initial graph of W. Piollet makes the convention that if a variable x and its inverse x-l do not occur in W, then loops are put at x and X1 and each forms a component of Xw. If c(W) is the number of connected components of & which have all vertices, or their inverses, in gen( W), then I_J?,l= c( W) + 2(d - /gen( W) I). Therefore, p(W)


I WI/z+ l-&l = lgen(W+4W+W-




Letting A ( W) = Igen( W) I - c(W), [3, Lemma 3.71 shows that if x is a W-attached regular H-map, then A( Wx) =fl( W). Furthermore, [3, Corollary 5.61 implies that c( Wx) = 1 if Wx is in canonical form. Thus we have, Igen(




= Igen(Wx)l -c(Wx) Therefore,

= Igen(Wx)I -1.

c(W) - 1 = Igen( W)] - Igen( Wx)/. It follows

P(W)-1 = 2d+C(W)-lgen(W>l - 1 = d_ 2 Thus



2 can be rewritten

that Igen( Wx)I 2 .


which is our version of Zieschang’s formula. Zieschang’s formula also follows easily from ours, in the case H= ( 1). Let W be a quadratic word without constants and let x be a W-attached H-automorphism such that Wx is in canonical form. There are two cases to consider. Case 1. If Wx = C, for some gl0, then x is a W-attached H-automorphism and Wx is of minimal singularity. Since the index sequence is empty, nullity(W) = singularity( Wx) = g + Igen(0) I = g.




C.C. Edmunds

Case 2. If Wx =S,, for some n>O, say wx =y:yi...yi for disjoint yi’s, let a. :y;+t H y;‘yi+, (i odd and i
= singularity(


Wa) =


L Since

Igen( Wx) 1= Igen( Wa)l , it follows nullity(W)



[ Thus,

by Proposition rank(W)

+1 2

Wa) =

1 .

in either case that


3.3, we have = d-

lgen(Wx)/ + 1 2

I *

7. Examples In conclusion we shall give two explain why a nullity formula and, more complex for equations with Consider the equation W= 1 in w2]x,

examples of quadratic equations which help to by Proposition 3.3, a rank formula is necessarily constants than for equations without constants. (u, u, W,X, y : 0) *(a, b : 0) with W equal to

Yl[a,bl b4 ulb, bl,

To compute the where [x,y] represents the commutator x-‘xy with xy=yplxy. singularity of W, we note that when W is factored as in Lemma 4.1, W= Uo. With gw= 0 and igen(U,)I = 5, it follows that singularity(W) = 5 and, by Theorem 5.2, the rank of W could be as small as 0. However, the regular W-attached H-map sends W to Wa = [x, y][u, 01w2[a, b12. a:x+bx w*, y H yw2, u H UW*[4b’rv++v ~2’Gb1 Since g wa = 2, k,, - 0, and U. = w2[a, b12 it follows that singularity( Wa) = 3. This shows that we must minimize over all such a to obtain the minimum singularity. It can be shown, in this case, that rank(W) = 2. In the previous example the rank is 2. This also happens to be the rank of Wrc ( = w2[x, y] [u, u]), by Zieschang’s formula. It is easy to see that rank(W) I rank( Wn) for any consistent quadratic word with constants. The following example shows that this inequality may be strict. Consider the equation W= 1 in F,*H where

It can be shown that rank(W) = 0. However WTC= 1; therefore rank( Wn) = d. Thus the increase in rank between W= 1 and W7c = 1 is d. Similar examples can be constructed showing that rank(Wrr) can take any value between rank(W) and d.


On the rank of quadratic equations

References [1] L.P.


Algebra [2] L.P.


(Singapore, [3] C.C.

and C.C.




over free groups

and free products,


68 (1981) 276-297. and C.C.


1987) (de Gruyter,



On the endomorphism


of equations

in free groups,

in: Group


1989) 241-251. problem

for free groups,

II, Proc.



Sot. 38

(1979) 153-168. [4] R.C.


The equation

[5] R.C.



[6] R.C.


and P.E.

a2b2=c2 in free groups, in groups,


[7] D. Piollet,



[8] D. Piollet,








J. 6 (1959) 89-95.

14 (1966) 275-283.


dans le groupe quadratique





J. Algebra

dans le groupe



33 (1975) 395-404.




59 (1986)

115-123. [9] A.A. Razborov, On systems [lo] H. Zieschang, Alternierende (1964) 13-31.

of equations in a free group, Math. U.S.S.R.-lzv. 25 (1985) 115-162. Produkte in freien Gruppen, Abh. Math. Sem. Univ. Hamburg 27