Mathematics

Ya. Ya. Sedol

THE FREE PRODUCT OF ASSOCIATIVE CALCULI WITH AMALGAMATED SUBALPHABET AND SOME RELATED QUESTIONS

(Presented by Academician P. S. Novikov on 27 IV 1964)

1. Let \(\Gamma = A \cup B\), \(H = A \cap B\), where \(A\) and \(B\) are alphabets. Let \(\mathfrak A\) and \(\mathfrak B\) be associative calculi\({}^{1}\), respectively, in the alphabets \(A\) and \(B\). We shall assume that for every word \(M\) in \(H\) there is a word \([M^{-1}]\) in \(H\) such that
   \[ \mathfrak A:\; M [M^{-1}M] \parallel [M^{-1}M] \parallel \Lambda \]
   (the group condition), and that for any words \(M\) and \(N\) in \(H\), from
   \[ \mathfrak A:\; M \parallel N \]
   there follows
   \[ \mathfrak B:\; M \parallel N \]
   and conversely (the isomorphism condition). Then the associative calculus \(\mathfrak G\) in the alphabet \(\Gamma\), determined by the system obtained by uniting the defining systems of the calculi \(\mathfrak A\) and \(\mathfrak B\), is called the free product of the associative calculi \(\mathfrak A\) and \(\mathfrak B\) with amalgamated subalphabet \(H\).

Theorem 1. Let \(\mathfrak G\) be the free product of the associative calculi \(\mathfrak A\) and \(\mathfrak B\) in the alphabets \(A\) and \(B\), respectively, with amalgamated subalphabet \(H\). Then, if \(P\) and \(Q\) are words in \(A\) and
\[ \mathfrak G:\; P \parallel Q, \]
then
\[ \mathfrak A:\; P \parallel Q. \]

Theorem 2. Under the same hypotheses, if \(P\) is a word in \(A\), \(Q\) is a word in \(B\), and
\[ \mathfrak G:\; P \parallel Q, \]
then one can find a word \(M\) in \(H\) such that
\[ \mathfrak A:\; P \parallel M \]
and
\[ \mathfrak B:\; Q \parallel M. \]

The method of proof is as follows. Introduce an auxiliary alphabet \(D=\{p,q\}\) consisting of two letters not belonging to the alphabet \(\Gamma\). A word \(W\) in the alphabet \(\Pi=\Gamma\cup D\) is called a proper word of order \(\nu\), if it has the form
\[ W=pK_1qL_1pK_2qL_2\ldots pK_\nu qL_\nu pK_{\nu+1}q, \]
where \(\nu \geq 0\); \(K_1,K_2,\ldots,K_{\nu+1}\) are words in \(A\); \(L_1,L_2,\ldots,L_\nu\) are words in \(B\). A proper word \(W\) is called perfect if none of the words
\[ L_1,\; K_2,\; L_2,\; \ldots,\; K_\nu,\; L_\nu \]
is a word in \(H\).

A sequence of \(2\nu+2\) numbers
\[ S=\{s_0,s_1,\ldots,s_{2\nu+1}\} \]
is called a proper numbering of order \(\nu\), if the following conditions are fulfilled.

a) The sequence \(S\) contains each of the numbers \(0,1,\ldots,\nu\) exactly twice.

b) If \(s_\alpha=s_\beta\), where \(0\leq \alpha<\beta\leq 2\nu+1\), then the number \(\beta-\alpha\) is odd.

c) There is no quadruple of numbers \(\alpha,\beta,\gamma,\delta\) such that
\[ 0\leq \alpha<\beta<\gamma<\delta\leq 2\nu+1 \]
and
\[ s_\alpha=s_\gamma,\qquad s_\beta=s_\delta. \]

d) \(s_0=s_{2\nu+1}=0\).

The following correspondence is introduced between the numbers of the numbering \(S\) and the occurrences of the letters of the alphabet \(D\) in a proper word \(W\): the number \(s_0\) is called the number of the first occurrence in \(W\) of the letter \(p\), \(s_1\) the number of the first occurrence in \(W\) of the letter \(q\), and so on. Occurrences having the same number constitute a pair, and the common number of these occurrences is called the number of the pair. The pair with number \(i\) is denoted by \(\omega_i\).

Let \(0\leq \alpha<\beta\leq 2\nu+1\) and \(s_\alpha=s_\beta=i\). The pair \(\omega_i\) is called a pair of type \(A\) (of type \(B\)) if \(\alpha\) is even (odd). The word \(\Omega_i\) is defined by putting, in the first case,
\[ \Omega_i=pK_{\frac{\alpha+2}{2}}qL_{\frac{\alpha+2}{2}}\ldots L_{\frac{\beta-1}{2}}pK_{\frac{\beta+1}{2}}q, \]

and in the second

\[ \Omega_i=qL_{\frac{\alpha+1}{2}}pK_{\frac{\alpha+3}{2}}\cdots K_{\frac{\beta}{2}}qL_{\frac{\beta}{2}}p. \]

The words \(\Omega_i\) \((i=0,1,\ldots,\nu)\) occur in the word \(W\), and \(\Omega_0=W\). Each of the words \(\Omega_i\) can be represented in a unique way in the form

\[ \Omega_i=pK_{t_1}\Omega_{u_1}L_{t_2}\cdots L_{t_r}\Omega_{u_r}L_{t_{r+1}}, \tag{1} \]

if \(\omega_i\) is a pair of type \(A\), or in the form

\[ \Omega_i=qL_{t_1}\Omega_{u_1}L_{t_2}\cdots L_{t_r}\Omega_{u_r}L_{t_{r+1}}, \tag{2} \]

if \(\omega_i\) is a pair of type \(B\). Here \(r\ge 0\); \(t_1,t_2,\ldots,t_{r+1}\) are some of the numbers \(1,2,\ldots,\nu+1\); \(u_1,u_2,\ldots,u_r\) are some of the numbers \(1,2,\ldots,\nu\).

Suppose we are given a list of words \(M_0,M_1,\ldots,M_\nu\), where \(M_0\) is a word in \(A\), and \(M_1,M_2,\ldots,M_\nu\) are words in \(H\). Construct the words \(N_0,N_1,\ldots,N_\nu\), putting, in accordance with (1) and (2),

\[ N_i=K_{t_1}M_{u_1}K_{t_2}\cdots K_{t_r}M_{u_r}K_{t_{r+1}}, \tag{3} \]

if \(\omega_i\) is a pair of type \(A\), or

\[ N_i=L_{t_1}M_{u_1}L_{t_2}\cdots L_{t_r}M_{u_r}L_{t_{r+1}}, \tag{4} \]

if \(\omega_i\) is a pair of type \(B\).

Let \(Q\) be a word in \(\Gamma\). A system of objects \(\{\nu,W,S,M_0,M_1,\ldots,M_\nu\}\), where \(\nu\) is a nonnegative integer, \(W\) is a proper word of order \(\nu\), \(S\) is a proper numbering of order \(\nu\), \(M_0\) is a word in \(A\), and \(M_1,M_2,\ldots,M_\nu\) are words in \(H\), is called an analysis of the word \(Q\) if the following conditions hold:

A.1. \([W]^{\Gamma}=Q\).

A.2. For all \(i=0,1,\ldots,\nu\), \(\mathfrak A: M_i\parallel N_i\), if \(\omega_i\) is a pair of type \(A\), or \(\mathfrak B: M_i\parallel N_i\), if \(\omega_i\) is a pair of type \(B\), where the words \(N_i\) are defined according to (3) and (4).

The word \(M_0\) is called the basis of the analysis. The analysis is called complete if the word \(W\) is complete.

Lemma 1. If \(Q\) is a word in \(\Gamma\), \(P\) is a word in \(A\), and \(\mathfrak G: P\parallel Q\), then the word \(Q\) has a complete analysis with basis \(P\).

The assertions of Theorems 1 and 2 are obtained from Lemma 1 as corollaries.

In the special case when \(\mathfrak A\) and \(\mathfrak B\) are inverse calculi \((^2)\), the results stated can be regarded as an introduction to the constructive theory of the free product of groups with an amalgamated subgroup.

1. Consider the case in which \(\mathfrak A\) and \(\mathfrak B\) are inverse calculi and the alphabet \(H\) is empty. Then the group conditions and the isomorphism condition are satisfied automatically, and the calculus \(\mathfrak G\) is called the free product of the inverse calculi \(\mathfrak A\) and \(\mathfrak B\). Let \(\mathfrak C=\{C_1,C_2,\ldots,C_m\}\) be some system of words in \(\Gamma\). A word \(Q\) is called dependent in \(\mathfrak G\) on \(\mathfrak C\) if

\[ \mathfrak G:\ Q\parallel C_{k_1}^{\varepsilon_1}C_{k_2}^{\varepsilon_2}\cdots C_{k_r}^{\varepsilon_r}, \]

where \(r=0\), \(1\le k_i\le m\), \(\varepsilon_i=\pm1\), \(i=1,2,\ldots,r\).

The system of words \(\mathfrak C\) is called a system of generators of the calculus \(\mathfrak G\) if every word in \(\Gamma\) is dependent in \(\mathfrak G\) on \(\mathfrak C\).

Theorem 3 (constructive form of Grushko’s theorem). Let \(\mathfrak G\) be the free product of inverse calculi \(\mathfrak A\) and \(\mathfrak B\) in the alphabets \(A\) and \(B\), respectively. Then, if the calculus \(\mathfrak G\) has a system of generators consisting of \(m\) words, one can find a system of generators of the calculus \(\mathfrak G\) consisting of no more than \(m\) words, each of which is a word in one of the alphabets \(A,B\).

A constructive proof of Theorem 3 was obtained by me on the basis of Lemma 1. Its plan was borrowed from Neumann’s paper \((^3)\).

1. Let \(\mathfrak A\) be an inverse calculus in some alphabet \(A\), and let

\[ Б=\{b_1,b_2,\ldots,b_p,b_1^{-1},b_2^{-1},\ldots,b_p^{-1}\} \]

be an alphabet having no letters in common with \(A\). Let \(\mathfrak G\) be an inverse calculus in the alphabet \(\Gamma=A\cup Б\), defined by the system

\[ \begin{cases} U_i \leftrightarrow\\ b_{v_j}^{-1}A_jb_{v_j}\mid B_j^{-1}\leftrightarrow \end{cases} \]

\[ (i=1,2,\ldots,n,\quad j=1,2,\ldots,m), \]

where \(\{U_i\leftrightarrow (i=1,2,\ldots,n)\}\) is the defining system of the calculus \(\mathfrak A\), \(1\le v_j\le p\), and \(A_j,B_j\) are words in \(A\). Suppose that the following isomorphism condition is satisfied:

If

\[ \mathfrak A:\ A_{k_1}^{\varepsilon_1}A_{k_2}^{\varepsilon_2}\cdots A_{k_r}^{\varepsilon_r}\parallel \Lambda, \]

then

\[ \mathfrak A:\ B_{k_1}^{\varepsilon_1}B_{k_2}^{\varepsilon_2}\cdots B_{k_r}^{\varepsilon_r}\parallel \Lambda \]

and conversely, where \(r\ge1\), \(\varepsilon_1,\varepsilon_2,\ldots,\varepsilon_r=\pm1\), and \(k_1,k_2,\ldots,k_r\) are some of the numbers \(1,2,\ldots,m\). Then the letters \(b_1,b_2,\ldots,b_p\) are called stable letters for the calculus \(\mathfrak G\) with base \(\mathfrak A\).

Theorem 4. Let the inverse calculus \(\mathfrak G\) in the alphabet \(\Gamma\) have stable letters \(b_1,b_2,\ldots,b_p\) with base \(\mathfrak A\) in the alphabet \(A\). Then, if \(Q\) is a word in \(A\) and

\[ \mathfrak G:\ Q\parallel \Lambda, \]

then

\[ \mathfrak A:\ Q\parallel \Lambda. \]

Theorem 5. Under the same assumptions, if \(Q\) is a word in \(\Gamma\) which is not a word in \(A\), and

\[ \mathfrak G:\ Q\parallel \Lambda, \]

then there is an occurrence in the word \(Q\) of some word of the form

\[ b_{v_j}^{\varepsilon}Cb_v^{-\varepsilon}, \]

where \(1\le v\le p\), and \(C\) is a word in \(A\) depending on the system of words \(\{A_1,A_2,\ldots,A_m\}\), if \(\varepsilon=-1\), or on the system \(\{B_1,B_2,\ldots,B_m\}\), if \(\varepsilon=1\).

For the proof of Theorems 4 and 5 one may apply a method analogous to the method used to prove Theorems 1 and 2.

1. Let \(A=\{a_1,a_2,\ldots,a_m\}\) be some alphabet, and let \(\mathfrak B\) be an associative calculus in the alphabet \(\overline{Б}=A\cup\{q\}\), defined by the system

\[ \{F_iq\leftrightarrow qK_i\}\quad (i=1,2,\ldots,n), \]

where \(F_i,K_i\) are words in \(A\). Words in \(\overline{Б}\) having exactly one occurrence of the letter \(q\) will be called special. Let \(\Gamma\) be the alphabet consisting of the letters

\[ a_1,a_2,\ldots,a_m,\ q,\ k,\ t,\ x,\ y,\ l_1,l_2,\ldots,l_n,\ r_1,r_2,\ldots,r_n \]

and their inverses. Let \(\mathfrak G\) be the inverse calculus in the alphabet \(\Gamma\), defined by the system

\[ \left\{ \begin{aligned} &yya_jy^{-1}a_j^{-1}\leftrightarrow\\ &a_jxxa_j^{-1}x^{-1}\leftrightarrow\\ &yl_iya_jl_i^{-1}a_j^{-1}\leftrightarrow\\ &a_jxr_ix a_j^{-1}r_i^{-1}\leftrightarrow\\ &l_iqK_ir_iq^{-1}\mid F_i^{-1}\leftrightarrow\\ &l_itl_i^{-1}t^{-1}\leftrightarrow\\ &yty^{-1}t^{-1}\leftrightarrow\\ &kr_ik^{-1}r_i^{-1}\leftrightarrow\\ &kxk^{-1}x^{-1}\leftrightarrow\\ &kq^{-1}tqk^{-1}q^{-1}t^{-1}q\leftrightarrow \end{aligned} \right. \]

\[ (i=1,2,\ldots,n;\quad j=1,2,\ldots,m). \]

Theorem 6. If \(\Sigma\) is a special word in \(\overline{Б}\), then from

\[ \mathfrak B:\ \Sigma\parallel q \]

it follows that

\[ \mathfrak G:\ k[\Sigma^{-1}t\Sigma k^{-1}[\Sigma^{-1}t^{-1}\Sigma\parallel \Lambda \]

and conversely.

Let \(\mathfrak A\) be an associative calculus in the alphabet \(A=\{a_1a_2,\ldots,a_m\}\), defined by the system

\[ \{T_i\leftrightarrow U_i\}\quad (i=1,2,\ldots,l), \]

with an undecidable equivalence problem for the empty word (such a calculus was constructed

by A. A. Markov ([1]). Let \(\mathfrak{B}\) be an associative calculus in the alphabet
\[ B=A\cup\{q\}, \]
defined by the system
\[ \begin{cases} T_i q \leftrightarrow q U_j,\\ a_j q \leftrightarrow q a_j \end{cases} \]
\[ (i=1,2,\ldots,l;\; j=1,2,\ldots,m). \]

Then, for the calculus \(\mathfrak{B}\), the problem of equivalence of an arbitrary special word to the word \(q\) is undecidable.

Now Theorem 6 implies:

Theorem 7. An inverse calculus \(\mathfrak{G}\) with an undecidable equivalence problem can be constructed.

The statements of Theorems 4–7, as well as the plan of the proofs of Theorems 6 and 7, are taken, with some modifications, from Britton’s paper ([4]). Thus a comparatively simple constructive proof has been obtained of P. S. Novikov’s result ([5]) on the “algorithmic undecidability of the identity problem” for finitely presented groups.

Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
2 IV 1964

References

[1] A. A. Markov, Trudy Mat. Inst. im. V. A. Steklova AN SSSR, 42 (1954).
[2] A. A. Markov, Izv. AN SSSR, Ser. Mat., 27, no. 4, 907 (1963).
[3] B. H. Neumann, J. London Math. Soc., 18, 12 (1943).
[4] J. L. Britton, Ann. Math., 77, 16 (1963).
[5] P. S. Novikov, DAN, 85, 709 (1952).