Conjugacy separability of certain torsion groups

It is shown that certain torsion groups studied by Grigorchuk and by Gupta and Sidki are conjugacy separable. 1. Introduction. In [1, 2, 3], R. I. Grigorchuk introduced and studied some remarkable new examples of finitely generated torsion groups. These groups are residually finite and amenable, and they fall into 20 isomorphism classes of p-groups for each prime p. Grigorchuk solved a problem of Milnor by showing that their growth is faster than polynomial growth and slower than exponential growth, and he gave in [2] an explicit criterion describing which of these groups have soluble word problem. However, the conjugacy problem for these groups seems to remain open. We are grateful to Professor Grigorchuk for pointing this out to us and for a number of helpful suggestions. We recall that a group G is conjugacy separable if for every pair g, h of elements of G which are not conjugate there is a finite quotient group of G in which the images of g, h fail to be conjugate; equivalently, G is conjugacy separable if (i) G is residually finite, so that it embeds naturally in its profinite completion b G, and (ii) elements of G which are conjugate in b G are also conjugate in G. A well-known argument of McKinsey [5] shows that if G is a finitely generated recursively presented group and if G is conjugacy separable then the conjugacy problem for G is soluble. We shall prove that for p odd the p-groups of Grigorchuk described above are conjugacy separable. It follows from this that if G is one of these p-groups then G has soluble conjugacy problem if and only if it has soluble word problem. Further remarkable examples of finitely generated residually finite p-torsion groups have been discovered and studied by N. Gupta and S. Sidki [4, 6], and we shall show that for p odd they too are conjugacy separable. Since these latter groups are recursively presented, it follows that they have soluble conjugacy problem. The groups mentioned above have a number of common features. Each of them can be embedded as a subgroup of finite index in the wreath product of a similar group and a finite cyclic group, and there is a length function which behaves well with respect to the embedding. We shall not give here the details of the constructions of these groups; instead we shall prove a result sufficiently general to handle all of them simultaneously. As a byproduct, we shall prove that the standard wreath product of a conjugacy separable group and a finite group is conjugacy separable. Arch. Math. 68 (1997) 441–449 0003-889X/97/060441-09 $ 3.30/0 © Birkhäuser Verlag, Basel, 1997 Archiv der Mathematik Unfortunately the 2-groups of Grigorchuk and of Gupta and Sidki are not covered by our result. Although they can probably be handled using similar techniques (especially the very explicitly described group in [1]), we believe that the arguments required would be considerably more complicated. This paper was written while the second author was visiting the University of Birmingham. He would like to thank the School of Mathematics and Statistics for its support and warm hospitality. 2. Wreath products and the main theorem. First we establish our notation for wreath products. Let H;K be groups, and write B for the group of all functions of finite support from K to H with pointwise multiplication. Let K act on the right on B as follows: b…x† ˆ b…x k† for k, x 2 K, b 2 B: The standard (restricted) wreath product W ˆ H wrK is the split extension of B by K, and B is the base group of W. If K is a finite cyclic group of order m generated by t we shall find it convenient to denote the element b 2 B by the vector …b…1†, b…t†, . . . , b…tmÿ1††; thus in this notation we have t…h1, . . . , hm†t ˆ …hm, h1, . . . , hmÿ1† for all …h1; . . . ;hm† 2 B. Lemma 1. Let g1 ˆ b1k, g2 ˆ b2k for some k 2 K, b1, b2 2 B and suppose that k has finite order m. If g 1 , g m 2 are conjugate in B then g b 1 ˆ g2 for some b 2 B. Proof . Suppose that …g 1 † d ˆ g 2 , where d 2 B. We have g 1 ˆ b1k b1k ˆ b1b kÿ1 1 b kÿ…mÿ1† 1 since km ˆ 1. Thus for each x 2 K we have g 1 …x† ˆ b1…x†b1…x k† b1…x k mÿ1 †: