Reports of the Academy of Sciences of the USSR

1. Volume 134, No. 1

MATHEMATICS

A. V. GLADKII

ON GROUPS WITH A (k)-REDUCIBLE BASIS

(Presented by Academician S. L. Sobolev, 18 IV 1960)

The groups with a (\delta)-basis and with a (k)-reducible basis introduced by V. A. Tartakovskii are of interest in view of the existence for them, respectively for (\delta < {}^{1}/_{6}) and (k > 6), of algorithms solving the identity problem*. The question arises of finding conditions, independent of the way in which the group is given, that are necessary or sufficient for the existence of a (k)-reducible basis with “good” (k)**. One such necessary condition is given in ((^{2})). In the present note other necessary conditions are given (Theorems 5 and 6), one of which absorbs the condition from ((^{2})). To obtain these conditions, Theorems 2, 3, and 4 are proved, which are also of independent interest. The question of the relationship between the notions of a (k)-reducible basis and a (\delta)-basis is also investigated (Theorem 1).

Let (G) be a finitely defined group with defining relations (A_i = 1) ((i = 1, \ldots, n)), where the words (A_i) are irreducible and externally irreducible (i.e. the first and last letters of (A_i) are not mutually inverse). The closure of the set ({A_i}) with respect to the operations of cyclic permutation of letters and free inversion will be called the basis of the group, and the words belonging to the basis will be called basis words.

Let (A) be a basis word and (B) a segment of a basis word, where
(A \doteq A'F,\ B \doteq F^{-1}B',\ F \ne \Lambda)***. The operation on a pair of words ({A,B}), the result of which is the pair of words ({A',B'}), will be called a left reduction of (B) by means of (A). A right reduction of (B) by means of (A) is defined analogously.

If (M,N) are basis words and (N \doteq PQR), then the left (right) reduction of (Q) by means of (M) will be called simple if (M^{-1} \ne QRP) ((M^{-1} \ne RPQ)). The simplicity of the reduction depends not only on (Q) and (M), but also on (N).

Further, if (N) is a basis word and (N \doteq RQR), then an operation consisting in the successive application to (Q) of (s) simple reductions (some of which may be left and some right) by means of some basis words will be called an (s)-fold simple reduction of (Q). If the word (Q) can be transformed into (\Lambda) by an (s)-fold simple reduction, we shall call it (s)-reducible.

Definition 1. Let (k) be a natural number. A basis is called (k)-reducible if: 1) for (t < k) no basis word is (t)-reducible; 2) there exists a (k)-reducible basis word; 3) every basis word is (s)-reducible for some (s).

* The first such algorithm was found by V. A. Tartakovskii ((^{16,\mathrm{в}})). Subsequently M. Greendlinger (in a work whose contents he presented at the seminar on mathematical logic at Moscow State University on 27 IV and 4 V 1958) indicated another, much simpler algorithm for groups with a (\delta)-basis. For some related classes of groups the identity problem was solved by Britton ((^{4})) and Shik ((^{3})).

** P. S. Novikov drew my attention to this question, to whom I express my deep gratitude.

*** (\doteq) denotes graphical equality, (F^{-1}) is the free inverse of (F), (\Lambda) is the empty word.

Definition 1′. A basis satisfying conditions 1) and 2) of Definition 1 is called a generalized (k)-reducible basis.

Definition 1″. A basis is called irreducible if no basis word is (s)-reducible for any (s).

We shall call the length a function (l(P)), defined on the set of words in some alphabet and satisfying the conditions:
1) (l(P)\geq 0); 2) (l(PQ)\leq l(P)+l(Q)).

Definition 2. Let a length (l) be defined on the set of words in the alphabet of a finitely defined group (G), and let (\delta>0) be a real number. A basis of the group is called a (\delta)-basis with respect to (l) if, for any basis words (M,N), from (M\not\equiv N^{-1}), (M\underset{\circ}{=}M'F), (N\underset{\circ}{=}F^{-1}N') it follows that
(l(M')\geq l(M)(1-\delta)), (l(N')\geq l(N)(1-\delta))*.

Theorem 1. Whatever the natural number (k), in a finitely defined group with a generalized (k)-reducible or irreducible basis one can define a length such that the basis with respect to this length will be a (\frac1k)-basis. Conversely, for (\delta<\frac1s), every (\delta)-basis (with respect to any length) is either generalized (k)-reducible for some (k>s), or irreducible.

In what follows we use the notions of composition and product introduced by V. A. Tartakovskii ((^{1a})).

Theorem 2. Every nonempty irreducible Dyck word of a finitely defined group (G) (i.e., a word equal to (1) in (G)) contains an occurrence of a nonempty word freely equal to a product of basis words.

A product of basis words is called simple if, under any method of reducing it, only simple reductions occur between its factors (cf. ((^{1a})), Ch. II, § 5).

Theorem 3. Every nonempty irreducible Dyck word of a finitely defined group (G) contains an occurrence of a nonempty word freely equal to a simple product of basis words.

Theorem 4. If (G) is a finitely defined group with a generalized (k)-reducible basis, where (k\geq 6), or with an irreducible basis, then for every nonempty irreducible Dyck word (P) of the group (G) there exist words (F,G,A,B) and a basis word (M) such that (P\underset{\circ}{=}AFB), (M\underset{\circ}{=}FG), and (G) is either empty or (i)-reducible, where (i\leq 3)*.

Theorem 5. A group with a (k)-reducible basis for (k\geq 6) cannot be periodic.

Theorem 6. In a group with a (k)-reducible basis for (k\geq 8), no nontrivial (i.e., not true in every group) identity relation can hold.

Theorems 5 and 6 are also true for groups with a generalized (k)-reducible basis and with an irreducible basis, except for the following trivial cases: 1) cyclic groups with generators (a,b_1,\ldots,b_n) and relations (ab_1^{\beta_1}=1,\ldots,ab_n^{\beta_n}=1), to which (a^\alpha=1) may be added; 2) the group with generators (a,b) and relations (a^2=1), (b^2=1) (in it the identity (x^2y^2x^{-2}y^{-2}=1) holds).

For groups with an (a)-reducible basis in the sense of V. A. Tartakovskii’s original definition, Theorems 5 and 6 are likewise valid if the formal orders of all generators (((^{1a}),) Ch. I, § 1) are infinite.

* Definitions 1 and 2 are not equivalent to the corresponding definitions of V. A. Tartakovskii, but are very close to them.

** This theorem generalizes the theorem from ((^{1a})), Ch. I, § 4, proved for the case when each generator occurs in some basis word.

*** When the present paper had been written, I became aware of the work ((^{4b})), where a theorem very close to Theorem 4 is proved. The class of groups considered in ((^{4b})) is close to the class of groups with a (\delta)-basis for (\delta<1/6), but neither contains the other.

If finite formal orders are present, these theorems can be proved with the lower bound for (k) increased respectively to 7 and to 10.

Let us also note the following generalization of Theorem 4:

Theorem (4'). Let every basis word (B_i) of the group (G) be represented in the form (B_i \doteq B_{i1}\ldots B_{is_i}), where (s_i \geqslant 6), and let every product (P) of basis factors of the group (G) be freely equal to such a product (Q) of basis factors (not necessarily the same ones) that, under any reduction of (Q), every simple cancellation between factors (B_i) and (B_j) annihilates no more than one (B_{ik_i}) and no more than one (B_{jk_j}) ((1 \leqslant k_i \leqslant s_i,\ 1 \leqslant k_j \leqslant s_j)). Then every Dick word of the group (G) contains a segment of some basis word (B_l) having the form (B_{lt}B_{l,t+1}B_{l,t+2}).

The assertion of Theorem (4') remains true if the arbitrary product (P) is replaced by a simple one, while requiring in addition that (Q) also be simple.

Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR

Received
3 III 1960

REFERENCES

1. V. A. Tartakovskii, a) Mat. sbornik, 25 (67), 1, 3 (1949); b) 25 (67), 2, 251 (1949); c) Izv. AN SSSR, ser. matem., 13, No. 6, 483 (1949).
2. A. V. Gladkii, DAN, 125, No. 5 (1959).
3. H. Schiek, Acta Math., 96, 157 (1956).
4. J. L. Britton, a) Proc. Glasgow Math. Assoc., 3, 1, 45 (1956); b) 3, 2, 68 (1957).