MATHEMATICS

MARTIN GREENDLINGER

SOLUTION OF THE CONJUGACY PROBLEM FOR ONE CLASS OF GROUPS COINCIDING WITH THEIR ANTICENTERS, BY MEANS OF A GENERALIZED DEHN ALGORITHM

(Presented by Academician A. I. Mal’cev, May 4, 1964)

Let a group \(G\) be given by generators \(a_1,\ldots,a_n\) and defining relations \(R_1=1,\ldots,R_k=1\), where: 1) each \(R_i\) is irreducible; 2) the set of words \(\{R_i\}\) is closed under the operations of taking the inverse word and taking cyclic permutations of the letters of the words \(R_i\); 3) if \(R_i\) and \(R_j\) are not mutually inverse, then, upon reduction of the product \(R_iR_j\), \(<\frac{1}{6}\) of the letters of the word \(R_i\) are cancelled.

An arbitrary word from the set \(\{R_i\}\) is called a defining word and is denoted by \(R\); the symbol \(\overline{T}\) denotes \(T^{-1}\); graphical equality, equality in the free group, and equality in the group \(G\) are denoted respectively by \(\simeq\), \(\equiv\), and \(=\); if \(\varepsilon_i=\pm 1\), then \(l(a_{i_1}^{\varepsilon_1}\ldots a_{i_m}^{\varepsilon_m})=m\)—the length of the word \(a_{i_1}^{\varepsilon_1}\ldots a_{i_m}^{\varepsilon_m}\).

Theorem 1. If an irreducible nonempty word \(W=1\) in the group \(G\), then \(W\) contains, without overlap:

1) \(>\frac{5}{6}\) of one \(R\),

or

2) \(>\frac{4}{6}\) of each of some two \(R\),

or

3) \(>\frac{4}{6}\) of one \(R\) and \(>\frac{3}{6}\) of each of some two \(R\),

or

4) \(>\frac{3}{6}\) of each of some five \(R\).

This theorem strengthens the main theorem from \((^6)\), but for a narrower class of groups. Its proof is based on the weaker main theorem from \((^5)\) and Lemmas 1–4. Analogous results for similar classes of groups are set forth in \((^1,^2,^4,^8,^{10-12})\).

Lemma 1. Let \(R_i\simeq S\overline{T}\) and \(XSY\) be irreducible.

1) If the subword \(S\) is not contained in any longer common subword of the words \(SY\) and \(R\), then \(TY\) is irreducible.

2) If \(T\simeq T^lT^r\), \(Y\simeq Y^lY^r\), and \(R_j\simeq TY^lV\), then \(l(T^l)<\frac{1}{6}l(R_i)\) and \(l(T^r)<\frac{1}{6}l(R_j)\).

Lemma 2. Let \(W\simeq XPQSY\), \(R_i\simeq PQ\overline{T}\), and \(R_j\simeq QS\overline{V}\).

1) If \(PQ\) is not contained in any longer common subword of the words \(W\) and \(R\), then \(l(Q)<\frac{1}{6}l(R_i)\) and \(l(Q)<\frac{1}{6}l(R_j)\).

2) If \(l(S)\geq \frac{1}{6}l(R_j)\) and \(T\simeq Ua\), then \(aSZ\) is not a defining word.

Lemma 3. If \(XS_1S_2Y\) is irreducible, \(R_i\simeq S_1\overline{T}_1\), \(R_j\simeq S_2\overline{T}_2\), \(T_1\simeq A_1P\), and \(T_2\simeq \overline{P}A_2\), then \(l(P)<\frac{1}{6}l(R_i)\) and \(l(P)<\frac{1}{6}l(R_j)\).

If, moreover, \(l(P)>0\), \(A_1\simeq B_1C_1\), \(A_2\simeq B_2C_2\), and \(R_k\simeq \overline{C}_1B_2Z\), then \(l(C_1)<\frac{1}{6}l(R_k)\) and \(l(B_2)<\frac{1}{6}l(R_k)\).

Lemma 4. If \(R_i \cong S_1\overline{T}_1\), \(R_j \cong S_2\overline{T}_2\), \(T_1 \cong AB\), \(S_2 \cong CD\), \(R_k \cong BCZ\), \(l(B)>0\), and \(T_1T_2\) is irreducible, then \(l(C)<\frac{1}{6}l(R_k)\).

Theorem 2. If an irreducible nonempty circular word \(C\) is equal to \(1\) in the group \(G\), then \(C\) either is a defining word written cyclically, or contains without intersection:

1) \(>\frac{5}{6}\) of each of some two \(R\),

or

2) \(>\frac{4}{6}\) of each of some three \(R\),

or

3) \(>\frac{4}{6}\) of each of some two \(R\) and \(>\frac{3}{6}\) of each of some two \(R\),

or

4) \(>\frac{4}{6}\) of one \(R\) and \(>\frac{3}{6}\) of each of some five \(R\),

or

5) \(>\frac{3}{6}\) of each of some seven \(R\).

This theorem strengthens Theorem 9 of \((^{6})\). Its proof is based on Theorem 1 and Lemmas 1–3. The author has constructed examples showing that Theorems 1 and 2 cannot be strengthened by increasing the fractions or the integers in any of the cases.

Definition 1. Let
\[ R_{i_1}\cong U_1V_1W_1,\qquad R_{i_2}\cong U_2\overline{W}_1V_2W_2,\qquad R_{i_3}\cong U_3\overline{W}_2V_3, \]
the words \(W_1\) and \(W_2\) being nonempty, \(l(V_2)>\frac{4}{6}l(R_{i_2})\), and either \(l(V_1)>\frac{2}{6}l(R_{i_1})\) and \(l(V_3)>\frac{3}{6}l(R_{i_3})\), or \(l(V_1)>\frac{3}{6}l(R_{i_1})\) and \(l(V_3)>\frac{2}{6}l(R_{i_3})\). Then the word \(V_1V_2V_3\) is called a triple, and \(V_2\) is called its core.

Theorem 3. Let an irreducible nonempty circular word \(C\) be equal to \(1\) in the group \(G\) and satisfy the following conditions:

a) \(C\) is not a defining word written cyclically;

b) \(C\) does not contain without intersection:

1) \(>\frac{5}{6}\) of each of some two \(R\),

or

2) \(>\frac{4}{6}\) of each of some three \(R\),

or

3) \(>\frac{3}{6}\) of each of some five \(R\).

If, in addition, \(R_i\cong S\overline{T}\), \(\frac{5}{6}l(R_i)\geq l(S)>\frac{4}{6}l(R_i)\), and the subword \(S\) is contained in \(C\), but is not contained in any longer common subword of the words \(R\) and \(C\), then \(S\) is the core of a triple that is a subword of \(C\).

In the proof of Theorem 3, Theorem 2 and Lemma 1 are used.

Definition 2. Let
\[ R_{i_1}\cong U_1V_1W_1,\qquad R_{i_2}\cong U_2\overline{W}_1V_2W_2,\qquad R_{i_3}\cong U_3\overline{W}_2V_3W_3,\qquad R_{i_4}\cong U_4\overline{W}_3V_4, \]
the words \(W_1\), \(W_2\), \(W_3\) being nonempty, \(l(V_1)>\frac{2}{6}l(R_{i_1})\), \(l(V_4)>\frac{2}{6}l(R_{i_4})\), and either

1) \(l(V_2)>\frac{4}{6}l(R_{i_2})\) and \(l(V_3)>\frac{3}{6}l(R_{i_3})\),

or

2) \(l(V_2)>\frac{3}{6}l(R_{i_2})\) and \(l(V_3)>\frac{4}{6}l(R_{i_3})\).

Then the word \(V_1V_2V_3V_4\) is called a quadruple, and \(V_2\) is its core in the first case, while \(V_3\) is its core in the second.

Theorem 4. Let an irreducible nonempty circular word \(C\) be equal to \(1\) in the group \(G\) and contain without intersection:

1) \(>\frac{5}{6}\) of one \(R\),

or

2) \(>\frac{4}{6}\) of each of some three \(R\),

or

3) \(>\frac{3}{6}\) of each of some five \(R\).

If, in addition, \(R_i\cong S\overline{T}\), \(\frac{5}{6}l(R_i)\geq l(S)>\frac{4}{6}l(R_i)\), and \(S\) is contained in \(C\), but is not contained in any longer common subword of the words \(R\) and \(C\), then \(S\) is the core of a quadruple that is a subword of \(C\).

The proof of Theorem 4 is based on Theorems 2 and 3 and Lemmas 1 and 3.

Theorem 5. If an irreducible nonempty circular word \(C\) is equal to \(1\) in the group \(G\), then either \(C\) contains the words \(S_j\) \((i=1,2,3,4)\) without intersection, where
\[ R_{i_j}\cong S_jT_j,\qquad l(S_1)>\frac{4}{6}l(R_{i_1}),\qquad l(S_2)>\frac{4}{6}l(R_{i_2}),\qquad l(S_3)>\frac{3}{6}l(R_{i_3}), \]

\(l(S_4) > \frac{3}{6} l(R_{i_4})\) and each of the words \(S_1\) and \(S_2\) is the core of a quadruple from \(C\), or \(C\) contains, without intersection:

1) \(> \frac{5}{6}\) of one \(R\),

or

2) \(> \frac{4}{6}\) of each of some three \(R\)’s,

or

3) \(> \frac{3}{6}\) of each of some five \(R\)’s.

In the proof of Theorem 5, Theorems 2 and 4 are used. Theorem 5, in turn, is applied to obtain the following two theorems, which constitute the main results.

Theorem 6. If \(XY = YX\) in the group \(G\), then \(X\) and \(Y\) are powers of one and the same element.

This was proved for generators \(X\) in \((^6)\), then for all \(X\) conjugate to words every cyclic permutation of which is irreducible and does not contain \(> \frac{1}{3}\) of a defining word, in \((^9)\), and for all elements of groups of one more restricted class in \((^7)\). The author has examples showing that the inequality \(< \frac{1}{6}\) is the best possible for Theorem 6.

The conjugacy problem for the group \(G\) is solved in the following way. In order to determine whether any two given words \(X\) and \(Y\) are conjugate, i.e., whether there exists a word \(Z\) such that \(X = \overline{Z}YZ\), write the word \(X\) on a circle; then reduce the resulting cyclic word and replace in it an occurrence of the word \(C\) by an occurrence of the word \(D\), if \(R_i \cong CD\) and \(l(C) > l(D)\). We continue this process as long as it is possible. Splitting the cyclic word transformed in this way at all possible places, we obtain a sequence of words \(A_1,\ldots,A_r\). If
\(R_i \cong \overline{B}A_jBC_j\), \(l(C_j) < \frac{2}{6}l(R_i) < l(A_j)\), and all cyclic permutations \(C_{j_1},\ldots,C_{j_s}\) of the word \(C_j\) are irreducible, then we replace the set \(\{A_i\}\) by the set \(\{C_{jk}\}\). Denote the words of the final list by \(X_1,\ldots,X_p\). 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
\[ l(Z) < \frac{2}{3}\max_{1\leq i\leq k} l(R_i), \]
and for each pair of words \((X_i,Y_j)\), we determine whether the equality
\[ X_i = \overline{Z}Y_jZ \]
holds.

Theorem 7. The words \(X\) and \(Y\) are conjugate if and only if at least one of the finite number of equalities
\[ X_i = \overline{Z}Y_jZ, \]
holds, where \(1 \leq i \leq p\), \(1 \leq j \leq q\), and
\[ l(Z) < \frac{2}{3}\max_{1\leq i\leq k} l(R_i). \]

This algorithm is a generalization of the algorithm which M. Dehn constructed for solving the conjugacy problem in another class of groups \((^3)\). Dehn’s original algorithm was applied in \((^6)\) to a narrower class of groups than the class of groups considered in the present paper. Another generalization of Dehn’s algorithm was applied in \((^8)\) to a class of groups that does not contain the class of groups considered here and is not contained in it.

Ivanovo State Pedagogical Institute
named after D. A. Furmanov

Received
29 IV 1964

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