Representations of Infinite Permutations by Words

We prove several cases of the following theorem: Every free group word which is not a proper power can represent every permutation of an infinite set. The remaining cases will be proved in a forthcoming paper of R. C. Lyndon. Fx denotes a free group freely generated by the set A. The elements of X are called letters, and the elements of Fx are represented by reduced words in those letters. G denotes an arbitrary group. We say that a word w can represent a in G ii a G G and there exists a homomorphism h: Fx —► G such that h{w) — a. In other words w can represent a in G iff the equation w — ais solvable in G, where X is the set of unknowns. Notice that, if is an endomorphism of Fx and {w) can represent a in G, then w can represent o in G. As was pointed out in [3], the sums of exponents of each letter of w have a greatest common divisior 1 iff w can represent every o in every G. This set of words with g.c.d. 1 is larger than the set of words w for which there exists an automorphism a oí Fx such that a{w) 6 X. E.g., by a theorem of J. H. C. Whitehead (see [5]), the words x2yx~1y~i and x2ys, for x,y £ X, are in the first set but not in the second set. The solvability of equations w = a for various groups G has been considered in several papers, see [2, 3, 4, 6, 7, 10]. In the present paper we will study this question for G = Sy, the group of all permutations of an infinite set Y. In [10] Silberger asks if every w € Fx which is not a power, i.e., is not of the form vk with k > 1, can represent every tt in 5y, if F is infinite. He shows that this is true if w — xmyn for x, y 6 A, x ^ y and m ^ 0 ^ n. Another proof is given in [2]. (For finite Y sufficient conditions in terms of m and n were given in [4]). Our main result is the following partial solution of Silberger's problem (it was announced in [8]). THEOREM 1. Ifw& Fx, w is not a power, it S Sy, and among the cycles of ■k at least one cycle size appears infinitely many times {fixed points are counted as cycles), then w can represent tt in Sy. REMARK ADDED IN MARCH 1986. R. C. Lyndon has just removed my assumption about the existence of repeating cycles in tt. Thus Silberger's conjecture is true. My proof uses some rather difficult tools of combinatorial group theory (a theorem of Weinbaum about subwords of a relator and the asphericity theorem for Cayley complexes of groups with one relator) but Lyndon's addition is more direct. First he points out that Theorem 1 reduces the problem to the cases when | y | ss N0 and either tt has at least one infinite cycle or it has no infinite cycles but Received by the editors June 12, 1985 and, in revised form, October 1985 and March 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 20F05; Secondary 20B07, 20E05. ©1987 American Mathematical Society 0002-9939/87 $1.00 + $.25 per page