UDC 519.41/47

MATHEMATICS

V. V. SOLDATOVA

ON A CLASS OF FINITELY PRESENTED GROUPS

(Presented by Academician P. S. Novikov on 23 IV 1966)

We shall denote by \(K\) the class of all finitely presented groups
\(G=\langle a_1,\ldots,a_n;\ R_1=1,\ldots,R_k=1\rangle\) such that: 1) the set of all defining words \(\{R_i\}\) is closed under the operation of taking inverses and cyclic permutation of letters; 2) each \(R_i\) is cyclically irreducible; 3) when reducing the product of two words \(R_{j_1}\) and \(R_{j_2}\) from \(\{R_i\}\) that are not mutually inverse, less than \(1/4\) of the letters of the word \(R_j\) are cancelled; 4) if the defining words \(R_{i_1}, R_{i_2}, R_{i_3}\) are written on a circle one after another and a reduction occurs between each pair, then less than \(1/4\) of each of the words \(R_{i_q}\), \(q=1,2,3\), is cancelled.

For groups of the class \(K\), the identity problem is solved by Dehn’s algorithm, and the conjugacy problem by the generalized Dehn algorithm.

In the work \((^1)\), M. Grindlinger gave positive solutions of the identity and conjugacy problems for a certain class of groups narrower than \(K\), and in the work \((^3)\) Lyndon did so for certain other classes, which partially intersect the class \(K\).

We shall use the following notation. If \(A\) is a word written in the generators of the group \(G\), then the inverse word to it will be denoted by \(\overline A\). By \(\equiv\), \(\cong\), \(=\) we shall denote, respectively, free equality, graphical equality, and equality in the group \(G\).

\(1^\circ\). In every group of the class \(K\), every word equal to the identity is freely equal to a product of \(n\) factors of the form \(\overline T_i R_i T_i\), where \(R_i\) is some defining word; \(T_i\) contains no more than half of some defining word, \(\overline T_i R_i T_i\) is irreducible; for any \(R_j\) and \(R_k\) from
\[ \prod_{i=1}^{n} \overline T_i R_i T_i \]
such that \(R_j\cong XYZ,\ R_k\cong P\overline YQ\), and under some way of reducing this product the words \(Y\) and \(\overline Y\) cancel with one another, we have \(ZXY\not\cong \overline YQP\).

The product \(R_{i_1}''\ldots R_{i_p}''\), remaining under some way of reducing
\[ \prod_{i=1}^{n}\overline T_i R_i T_i, \]
we shall call an adjacent product if
\[ \begin{gathered} \text{for } p=1\quad l(R_{i_1}'')> {}^3/{}_4\, l(R_{i_1}),\\ \text{for } p>1\quad l(R_{i_q}'')> {}^1/{}_2\, l(R_{i_q}),\quad q=1,p,\\ l(R_{i_j}'')> {}^1/{}_4\, l(R_{i_j}),\quad j=2,\ldots,p-1, \end{gathered} \]
each \(R_{i_q}\) is reduced with \(R_{i_{q+1}}\), \(q=1,\ldots,p-1\), and no \(R_{i_s}\) is reduced with any \(\overline T_{i_q}\) or \(T_{i_r}\).

Theorem 1. If a nonempty irreducible word \(C\) is equal to the identity in a group \(G\) from \(K\), then under any way of reducing the product
\[ \prod_{i=1}^{n}\overline T_i R_i T_i \equiv C, \]
the word \(C\) contains an adjacent product.

It is easy to see that Theorem 1 implies a positive solution of the identity problem for groups of the class \(K\).

\(2^\circ\). For every group in \(K\), denote by \(\mathcal A_1\) the set of all defining words \(R_i\); of the parts of the defining words \(R_i'\) remaining after reduction of the product of two not mutually inverse defining words; and of the parts \(R_i''\) of defining words remaining after reduction of \(R_i, R_j, R_k\), when they are written one after another on a circle and a reduction occurs between each pair.

Let the sets \(\mathcal A_2,\ldots,\mathcal A_{n-1}\) be defined. By \(\mathcal A_n\) we shall denote the set of all words freely equal to a product \(XY\) such that
\[ X \cong R_{i_1}^* \ldots R_{i_{n-1}}^* \in \mathcal A_{n-1}, \quad Y \cong R_j^* \in \mathcal A_1, \]
\(R_{i_{n-1}}^*\) and \(R_j^*\) are not parts of mutually inverse defining words, and \(R_{i_{n-1}}^* R_j^*\) is reducible.

The set
\[ \bigcup_{n=1}^{\infty} \mathcal A_n \]
will be denoted by \(\mathcal A\).

Lemma. A nonempty irreducible cyclic word equal to \(1\), not a defining word, and folded into a cycle, contains an occurrence of two nonintersecting words from the set \(\mathcal A\).

Let words \(A\) and \(B\) be given. In order to determine whether these words are conjugate or not, we fold each of them into a cycle, reduce them, and, if it contains a word \(S\) whose length is more than half of a defining word \(R \cong ST\), replace it by the smaller part \(\overline T\). If the word \(A\) transformed in this way is nonempty, we consider all its cyclic permutations \(A_1,\ldots,A_s\). If there exists a defining word \(R_i \cong X A_i \overline X D\) and \(l(D) < {}^1\!/_2 l(R_i)\), then we add to the sequence \(A_1,\ldots,A_s\) all possible permutations of \(D\). We proceed analogously with the word \(B\). We obtain two sequences \(A_1,\ldots,A_\alpha\) and \(B_1,\ldots,B_\beta\).

Theorem 2. The words \(A\) and \(B\) are conjugate if and only if at least one of the relations \(A_i = \overline X B_j X\) holds, where \(1 \leq i \leq \alpha\), \(1 \leq j \leq \beta\), and \(l(X) \leq {}^1\!/_2 \max\{l(R_i)\}\).

\(3^\circ\). In the following theorems a number of abstract properties of groups of the class \(K\) are established. In the proofs of these theorems the lemma given above is used.

Theorem 3. In every group of \(K\) which is not cyclic, no nontrivial identity holds.

Theorem 4. Every noncyclic group of \(K\) is a group without a center.

Theorem 5. Every element of a group in \(K\) has trivial centralizer.

I take this opportunity to express my gratitude to M. D. Grindlinger for his attention to the work and for valuable advice.

Ivanovo State Pedagogical Institute
named after D. A. Furmanov

Received
23 IV 1966

REFERENCES

1. M. Grindlinger, Izv. Akad. Nauk SSSR, Ser. Mat., 29, No. 2, 245 (1965).
2. M. Grindlinger, DAN, 154, No. 3, 507 (1964).
3. R. C. Lyndon, Notices Am. Math. Soc., 12, No. 3, 370 (1965).