Growth Series of Some Wreath Products

The growth series of certain finitely generated groups which are wreath products are investigated. These growth series are intimately related to the traveling salesman problem on certain graphs. A large class of these growth series is shown to consist of irrational algebraic functions. 0. Introduction Let r be a group with a finite generating set S. Define the S'-length \\g\\s of an element g in Y to be the least nonnegative integer n such that g can be expressed as a product of n elements from S U S~l. For every nonnegative integer n let a„ be the number of elements in Y with S'-length n. In particular, a0 = 1 (the identity element), and ax is the number of nonidentity elements in S U S~ ' . The growth series of the pair (Y, S) is by definition oo Mx) = zZ anX" • n=0 Since only one generating set will be associated with each group below, the generating set associated with fr(x) will be obvious. Let H be a group with a finite generating set Sh ■ These will be fixed for the rest of the paper. The Cayley graph of the pair (H, Sh) is, as usual, the directed graph whose vertices are the elements of H and there is an edge from a vertex hx Xoa vertex h2 if and only if h2 = hxh for some h in SH US^1. In particular, the edge from hx to h2 has an opposite edge from h2 to hx . Let C be the graph gotten from the Cayley graph of (H, Sh) simply by identifying opposite edges. In other words, C might be called the undirected Cayley graph of (H,SH). hex K also be a group with a finite generating set Sk ■ These will also be fixed for the rest of the paper. It is possible to form what might be called a restricted direct product group P, which consists of all functions p from the vertices of C to K such that there are only finitely many vertices v in C with p(v) t¿ 1. This is a subgroup of the direct product group whose elements consist of all functions from the vertex set of C to K. The group P admits H as a group of automorphisms by means of the action of H on C. The resulting semidirect product P x H is the restricted wreath product K\H. Received by the editors March 10, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 20F32, 05C25; Secondary 20E22. © 1992 American Mathematical Society 0002-9947/92 $1.00+ $.25 per page