MATHEMATICS

A. A. LORENTS

COEFFICIENT-FREE EQUATIONS IN FREE GROUPS

(Presented by Academician P. S. Novikov, 17 VII 1964)

In the present note we consider properties of “coefficient-free equations” in free groups, the precise definition of which is given below. Theorem 1 establishes for free groups essentially the same result as was obtained for equations in words by the Bulgarian mathematicians D. Skordev and Bl. Sendov (¹). In addition, the note considers systems of coefficient-free equations and gives a method for effectively describing the set of solutions for certain systems of a special form. Along with new notions we shall use the terms introduced by the author in (²).

Let a free group \(X\) with generators \(X_1, X_2, \ldots, X_n\) be given. We shall call expressions of the form

\[ W(X_1, X_2, \ldots, X_n)=1 \tag{1} \]

coefficient-free equations in the unknowns \(X_1, X_2, \ldots, X_n\), if \(W(X_1, X_2, \ldots, X_n)\) is an element of the group \(X\) such that for every \(i\) \((i=1,2,\ldots,n)\) either \(X_i\) or \(X_i^{-1}\) occurs in \(W(X_1, X_2, \ldots, X_n)\).

We shall call equation (1) normal if the element \(W(X_1, X_2, \ldots, X_n)\) is cyclically reduced.

Let a free group \(\mathfrak{G}\) of rank \(r\) and a sequence of elements of this group \(S=\{S_1,S_2,\ldots,S_n\}\) be given. Denote by \(W(S_1,S_2,\ldots,S_n)\) the element of the free group \(\mathfrak{G}\) obtained by substituting the terms \(S_i\) of the sequence \(S\) in place of \(X_i\) in \(W(X_1,X_2,\ldots,X_n)\).

It is natural to call a solution of equation (1) in the free group \(\mathfrak{G}\) any sequence \(S\) of elements of the free group \(\mathfrak{G}\) such that \(W(S_1,S_2,\ldots,S_n)=1\)*.

Definition 1. Let sequences \(S=\{S_1,S_2,\ldots,S_n\}\), \(T=\{T_1,T_2,\ldots,T_n\}\) of elements of a free group \(\mathfrak{G}\) be given. We define inductively the relation of association between \(S\) and \(T\):

1. \(S\) associates \(T\) if \(T_j=S_j^{e}S_k^{\varepsilon}\) \((e=\pm1,\ \varepsilon=\pm1,\ 1\le j,k\le n,\ j\ne k)\) and for all \(i\) \((i=1,2,\ldots,n)\), where \(i\ne j\), \(T_i=S_i\).

2. If the sequence \(S\) associates the sequence \(S'\) and \(S'\) associates the sequence \(T\), then \(S\) associates \(T\).

Definition 2. Suppose that free groups \(\mathfrak{G}_1\) of rank \(r_1\) and \(\mathfrak{G}_2\) of rank \(r_2\) \((r_2<r_1)\) are given. Fix the sequence

\[ S=\{S_1,S_2,\ldots,S_{r_2}\} \tag{2} \]

of elements of the group \(\mathfrak{G}\).

By the symbol \(\Phi_S\) we shall denote the mapping of the group \(\mathfrak{G}_2\) into \(\mathfrak{G}_1\) produced according to the following rules:

1. The image in \(\mathfrak{G}_1\) of a generator \(G_{2i}\) \((i=1,2,\ldots,r_2)\) of the group \(\mathfrak{G}_2\) is the term \(S_i\) from \(S\).

2. Let \(U=VG_{2i}^{e}\) \((e=\pm1)\) be an element of the group \(\mathfrak{G}_2\), and suppose that the image of the element \(V\) in \(\mathfrak{G}_1\) is the element \(V'\); then the image of \(U\) in \(\mathfrak{G}_1\) is the element \(U=V'S_i^{e}\).

* The identity element of the group \(\mathfrak{G}\) is denoted by \(1\).

The mapping \(\Phi_S\) of the group \(\mathfrak G_2\) into \(\mathfrak G_1\) is obviously a homomorphism. Consequently, the following proposition holds: if \(T\) is a solution of equation (1) in \(\mathfrak G_2\), then the image \(T'\) of the sequence \(T\) under the mapping \(\Phi_S\) is a solution of (1) in \(\mathfrak G_1\).

Theorem 1. Let a normal equation be given
\[ W(X_1, X_2,\ldots, X_n)=1, \tag{3} \]
a free group \(\mathfrak G\) of rank \(r \ge n\), a solution \(T\) of equation (3) in \(\mathfrak G\), and a free group \(\mathfrak G'\) of rank \(r'=n-1\). Then, whatever the sequence \(T\) may be, one can always indicate a solution \(S'\) of equation (3) in \(\mathfrak G'\) and a mapping \(\Phi_S\) of the free group \(\mathfrak G'\) into \(\mathfrak G\) such that the image of \(S'\) under the mapping \(\Phi_S\) is \(T\).

The proof of this theorem follows very simply from Lemmas 1 and 2.

Lemma 1. If \(T=\{T_1,T_2,\ldots,T_n\}\) is a solution of equation (3) in \(\mathfrak G\), and for every \(i\), \(T_i\ne 1\), then there exists a sequence \(S=\{S_1,S_2,\ldots,S_n\}\) such that \(T\) associates \(S\) and
\[ \sum_i |T_i^\delta| > \sum_i |S_i^\delta|. \]

Lemma 2. Let a sequence \(U=\{U_1,U_2,\ldots,U_m\}\) of elements of the group \(X\) and numbers \(e,\varepsilon\) satisfying the relations
\[ e=\pm1,\qquad \varepsilon=\pm1. \]
be given.

Suppose that \(U\) contains a pair of terms \(U_i, U_j\) such that \(U_i\ne U_j^{\pm1}\). Define another sequence \(U'=\{U_1',U_2',\ldots,U_m'\}\) in the same group \(X\) as follows:

1. \(U_k'=U_k\), if \(U_k\ne U_i^{e'}\) \((e'=\pm1)\).
2. \(U_k'=(U_i^e U_j^\varepsilon)^{e'}\), if \(U_k=U_i^{e'}\) \((k=1,2,\ldots,m)\).

If
\[ [U_1^\delta=[U_2^\delta=\cdots=[U_m^\delta=1, \]
\[ \prod_i U_i = U_1\cdot U_2\cdots U_m, \]
then
\[ \prod_i U_i' \ne 1. \]

A simple proof of Theorem 1 is obtained if, instead of Lemma 1, one uses in its expanded form Nielsen’s theorem that every subgroup of a free group of rank \(r\), generated by elements \(A_1,A_2,\ldots,A_l\) from \(\mathfrak G\), is isomorphic to some free group \(\mathfrak G^*\) with generators \(G_1^*,G_2^*,\ldots,G_{l'}^*\), where \(l'\le l\) \((^3)\). However, proving Lemma 1 is considerably simpler than proving Nielsen’s theorem.

We shall next consider coefficient-free equations of the form
\[ W(X,Y)=1, \tag{4} \]
i.e. equations in only two unknowns. Moreover, we shall suppose that every equation of the form (4) considered is normal and satisfies the condition
\[ W(X,Y)=\prod_k (X^{c_k}Y^{d_k}), \tag{5} \]
where \(c_k d_k\ne 0\).

From Theorem 1 it is not hard to derive the following result.

Corollary 1. A sequence \(S=\{S_1,S_2\}\) of elements of the free group \(\mathfrak G\) of rank \(r\) is a solution of equation (4) in \(\mathfrak G\) if and only if \(S_1=A^m,\ S_1=A^n\), where \(m,n\) are a solution of equation (6); \(A\) is an arbitrary element in \(\mathfrak G\).

The proposition just formulated makes it possible to prove a more general result, whose formulation requires additional terms.

Let there be given a system of coefficient-free equations \(\Omega\) in the unknowns \(X_1, X_2, \ldots, X_n\) such that each equation of the system \(\Omega\) has the form

\[ W(X_i, X_j)=1 \quad (i \ne j,\; 1 \le i,j \le n). \]

We shall henceforth denote by \(\sigma\) an arbitrary list of unknowns of the system \(\Omega\), assuming, moreover, that \(\sigma\) need not contain every unknown of the system \(\Omega\); in other words, the list \(\sigma\) need not be complete. In what follows we shall speak only of such lists \(\sigma\) that contain more than one unknown.

Definition 3. Fix a list \(\sigma\) of unknowns of the system \(\Omega\) and define a relation \(R\) for the unknowns of the list \(\sigma\) as follows:

1. \(X_i\) is in the relation \(R\) to \(X_i\) for every \(X_i\) in \(\sigma\) (reflexivity).

2. \(X_i\) is in the relation \(R\) to \(X_j\) if there is a sequence of unknowns \(X_{j_1}, X_{j_2}, \ldots, X_{j_l}\) of the list \(\sigma\) such that \(X_{j_1}=X_i\), \(X_{j_l}=X_j\), and for every \(k\), \(1 \le k \le l\), there is in the system \(\Omega\) an equation in the unknowns \(X_{j_k}, X_{j_{k+1}}\) \((j_k \ne j_{k+1})\).

The relation \(R\), obviously, determines a partition of the unknowns of the list \(\sigma\) into classes, since it is reflexive, symmetric, and transitive.

Definition 4. Let \(a_i\) be integers, \(t_i\) integer variables, and \(\Gamma_i\) variables with values in a free group \(G\) of finite rank. We shall call a sequence of expressions

\[ \Gamma_1^{a_1 t_1},\ \Gamma_2^{a_2 t_2},\ \ldots,\ \Gamma_n^{a_n t_n} \]

a biparametric sequence of type \(\sigma\), if the following conditions are satisfied:

1. \(a_i=0\) if the unknown \(X_i\) of the system \(\Omega\) does not belong to the list \(\sigma\).

2. In the case when the unknowns \(X_i\) and \(X_j\) belong to the list \(\sigma\), the variables denoted by the letters \(t_i\) and \(t_j\) coincide if and only if \(X_i\) is in the relation \(R\) to \(X_j\).

3. In the case when the unknowns \(X_i\) and \(X_j\) belong to the list \(\sigma\), the variables denoted by the letters \(\Gamma_i\) and \(\Gamma_j\) coincide if and only if \(X_i\) is in the relation \(R\) to \(X_j\).

We shall call a sequence \(S=\{S_1,S_2,\ldots,S_n\}\) of elements of the free group \(G\) a value of the biparametric sequence

\[ \Gamma_1^{a_1 t_1},\ \Gamma_2^{a_2 t_2},\ \ldots,\ \Gamma_n^{a_n t_n} \]

of type \(\sigma\), if there are sets of values of the variables \(\Gamma_i\) and \(t_i\) \((i=1,2,\ldots,n)\) such that, for every \(i\) \((i=1,2,\ldots,n)\),

\[ S_i=C_i^{a_i m_i}, \]

where \(C_i\) and \(m_i\) are the values of the variables \(\Gamma_i\) and \(t_i\) from these sets.

Theorem 2. For every system \(\Omega\) and every free group \(G\), one can construct a finite number of biparametric sequences of different types \(\sigma\) such that a sequence \(T\) of elements of the group \(G\) satisfies \(\Omega\) if and only if \(T\) is a value of one of these biparametric sequences.

Theorem 2 also holds for systems of equations in words, when the concepts “biparametric sequence of type \(\sigma\)” and “system of equations \(\Omega\)” are replaced by the corresponding equivalents.

Institute of Electronics and Computer Engineering
Academy of Sciences of the Latvian SSR

Received
3 January 1964

CITED LITERATURE

1. D. Skordew, B. E. Sendow, Zs. math. Logik u. Grundl. Math., 7, No. 4, 289 (1961).
2. A. A. Lorents, DAN, 148, No. 6, 1253 (1963).
3. M. Hall, Theory of Groups, IL, 1962.