MARTIN GREENDLINGER

SOLUTIONS OF THE WORD PROBLEM FOR ONE CLASS OF GROUPS BY MEANS OF DEHN’S ALGORITHM, AND OF THE CONJUGACY PROBLEM BY MEANS OF ONE GENERALIZATION OF DEHN’S ALGORITHM

(Presented by Academician P. S. Novikov on 30 IX 1963)

§ 1. Let a group \(G\) be given by generating elements \(a_1,\ldots,a_n\) and defining relations \(R_1=1,\ldots,R_k=1\), where: 1) each word \(R_i\) is reduced; 2) the set of words \(\{R_i\}\) is closed under the operations of taking inverses and taking cyclic permutations of the letters of the words \(R_i\); 3) if \(R_i\) and \(R_j\) are not mutual inverses, then \(<\frac14\) of the letters of the word \(R_i\) are cancelled in reducing the product \(R_iR_j\); 4) if each of the words \(R_i,R_j\), and \(R_k\) is written on one side of a triangle, then cancellation cannot occur at all three vertices.

The algorithm solving the word problem for the group \(G\) is as follows.

An arbitrary word \(W\) is given. In order to find out whether \(W\) is equal to the identity, we apply to the word \(W\), as long as this is possible, the following two operations: \(\alpha)\) reduction; \(\beta)\) replacement of \(S\) by \(T\), if \(R_i \simeq S\overline{T}\) and \(l(S)>l(T)\). (Here and below \(\simeq\), \(=\), and \(\equiv\) denote, respectively, graphical equality, equality in the free group, and equality in the group \(G\); \(l(a_{i_1}^{\varepsilon_1}a_{i_2}^{\varepsilon_2}\cdots a_{i_n}^{\varepsilon_n})=n\), and \(\overline{T}\) denotes \(T^{-1}\).)

By virtue of Theorem 2, \(W=1\) if and only if this process ends with the empty word. This algorithm was first applied by M. Dehn to another class of groups \((^1)\).

The conjugacy problem for the group \(G\) is solved in the following way. Let arbitrary two words \(X\) and \(Y\) be given. To find out whether they are conjugate, i.e. whether there exists a word \(Z\) such that \(X=\overline{Z}YZ\), we write the word \(X\) on a circle and apply, as long as this is possible, the operations \(\alpha)\) and \(\beta)\). Cutting the transformed circular word at all possible places, we obtain a sequence of words \(X_1,\ldots,X_j,\ldots,X_\alpha\). If \(R_i\simeq \overline{A}X_jA\overline{Q}_j\), \(l(Q_j)\leq \frac12 l(R_i)\), and all cyclic permutations \(Q_{j1},\ldots,Q_{j\beta}\) of the word \(Q_j\) are reduced, then we add them to the list of words \(X_1,\ldots,X_\alpha\). Then, with the help of the word \(Q_{jk}\), we obtain new words in the list in the same way as from the word \(X_j\), and so on.

Let us call the words in the final list \(X_1,\ldots,X_p\). From the word \(Y\) we obtain in an analogous way a list of words \(Y_1,\ldots,Y_q\). For each word \(Z\) such that \(l(Z)\leq \frac12\max_{1\leq i\leq k} l(R_i)\), and for each pair of words \((X_i,Y_j)\), we determine whether the equality
\[ X_i=\overline{Z}Y_jZ \]
holds.

Theorem 1. \(X\) and \(Y\) are conjugate if and only if at least one of this finite number of equalities holds.

This algorithm is a generalization of the algorithm which M. Dehn invented for solving the conjugacy problem for another class of groups \((^1)\).

§ 2. In order to formulate Theorem 2, it is first necessary to define the notion of an “adjacent \(n\)-gon.”

Definition 1. If, under the given method of cancellation (complete or partial) of the word $\prod_{i=1}^{m}\overline{T}_i R_iT_i$, the words $R_{ij}$ cancel with the words $R_{i_{j+1}}$
$(j=1,2,\ldots,n-1;\ 1\leqslant i_1<i_2<\cdots<i_n\leqslant m)$, no $R_{i_p}$ $(1\leqslant p\leqslant n)$ cancels with any $\overline{T}_i$ or $T_i$, except for the indicated cancellations, cancels with no more than one $R_i$ among one or two $R_{iq},R_{ip}$, and if, after cancellation, subwords $R_{i_p}^{c}$ $(1\leqslant p\leqslant n)$ remain from the words $R_{i_p}$, then the word $R_{i_1}^{c}R_{i_2}^{c}\ldots R_{i_n}^{c}$ is called an adjacent $n$-tuple.

Theorem 2. If an irreducible nonempty word $W$ is equal to the identity in the group $G$, then there exists a word $\prod_{i=1}^{m}\overline{T}_iR_iT_i$ such that
\[ W\equiv \prod_{i=1}^{m}\overline{T}_iR_iT_i \]
and, for any method of cancellation of the word $\prod_{i=1}^{m}\overline{T}_iR_iT_i$, an adjacent $n$-tuple remains.

To prove Theorem 2, products $\prod_{i=1}^{m}\overline{T}_iR_iT_i$ are considered such that $R_i$ absorbs $<\frac14$ of the letters of the word $R_j$ for all $i$ and $j$, and $\leqslant 2R_i$ cancel with $R_j$ on each side under any method of cancellation; each word $\overline{T}_iR_iT_i$ is irreducible, and neither $\overline{T}_i$ nor $T_i$ contains $>\frac12 R_j$.

For such products the following notation is introduced for infinite sequences of assertions.

$A_q$. If $m\leqslant q$ and $j<k$, then $\overline{T}_j$ does not cancel with $R_k$, and $T_k$ does not cancel with $R_j$, under any method of cancellation of the word $\prod_{i=1}^{m}\overline{T}_iR_iT_i$.

$B_q$. If $m\leqslant q$ and $j<k<p$, then $T_j$ and $T_k$ do not cancel with $R_p$, and $\overline{T}_k$ and $\overline{T}_p$ do not cancel with $R_j$ under any method of cancellation of the word $\prod_{i=1}^{m}\overline{T}_iR_iT_i$.

$C_q$. If $m\leqslant q$, $j<k<p$ (respectively, $j>k>p$), and $R_j$ cancels with $R_k$ and with $R_p$ under some method of cancellation of the word $\prod_{i=1}^{m}\overline{T}_iR_iT_i$, then $T_j$ (respectively, $\overline{T}_j$) also cancels with $R_k$.

$D_q$. If $m\leqslant q$, then under any method of cancellation of the word $\prod_{i=1}^{m}\overline{T}_iR_iT_i$ an adjacent $n$-tuple remains.

Lemma 1. $(A_q\ \&\ B_q\ \&\ C_q)\to B_{q+1}$.

Lemma 2. $(A_q\ \&\ B_{q+1})\to A_{q+1}$.

Lemma 3. $D_q\to C_{q+1}$.

Lemma 4. $(A_{q+1}\ \&\ B_{q+1}\ \&\ C_{q+1})\to D_{q+1}$.

For the proof of Lemmas 1 and 4 the following lemma is applied, valid for any product $\prod_{i=1}^{m}\overline{T}_iR_iT_i$ in any group satisfying property 4):

Lemma 5. If $i<j<k$, then it is impossible that $T_j$ cancels with $R_k$, $\overline{T}_j$ cancels with $R_i$, and $R_j$ cancels both with $R_i$ and with $R_k$.

Now, on the basis of mathematical induction, $A_q$, $B_q$, $C_q$, and $D_q$ are true for all natural numbers $q$. We easily get rid of the assumption that $\leqslant 2R_i$ cancel with each $R_j$ on each side, and Theorem 2 follows from Theorem 1 of paper $(^2)$.

§ 3. For the proof of Theorem 1 it is necessary to define the following sets:

\[ M_1=\{\text{all } R_i \text{ and all } R_j' \text{ such that } R_j \cong R_j'R_j'',\ R_k \cong \overline{R_j''}R_k',\ \overline{R_j}\not\equiv R_k\}. \]

Suppose that \(M_n\) has already been defined.

\[ \begin{aligned} M_{n+1}=\{&\text{all } R_{i_1}'\ldots R_{i_n}'R_{i_{n+1}}' \text{ such that } R_{i_j}\cong R_{i_j}'R_{i_j}''\ (1\leq j\leq n-1),\\ & R_{i_n}\cong R_{i_n}'''R_{i_n}'R_{i_n}'',\quad R_{i_{n+1}}\cong \overline{R_{i_n}''}R_{i_{n+1}}'R_{i_{n+1}}'',\\ & R_{i_1}'\ldots R_{i_{n-1}}'R_{i_n}'R_{i_n}''' \in M_n,\quad \overline{R_{i_n}''}R_{i_{n+1}}'\in M_1,\quad \overline{R_{i_n}}\not\equiv R_{i_{n+1}}\}. \end{aligned} \]

\[ M=\bigcup_{i=1}^{\infty} M_i. \]

Lemma 6. If a cyclically irreducible nonempty word \(W=1\) in the group \(G\), and \(W\) is written on a circumference, then the resulting circular word \(C\) contains two nonintersecting words from the set \(M\).

The fact that \(C\) contains two words from \(M\) follows directly from Theorem 2, since every adjacent \(n\)-tuple belongs to the set \(M_n\). The proof of nonintersection presents certain technical difficulties.

Theorem 1 is proved with the help of Lemma 6 and the definition of the group \(G\).

Received
17 IX 1963

REFERENCES

¹ M. Dehn, Math. Ann., 72, 413 (1912). ² M. Greendlinger, Comm. Pure and Appl. Math., 13, 641 (1960).