Growth series of some wreath products. (English) Zbl 0793.20034

Parry, Walter

Trans. Am. Math. Soc. 331, No. 2, 751-759 (1992).

Let \(G\) be a group with finite generating set. The growth power series of \(G\) relative to a generating set \(S\) is defined to be \(f_ G(x) = \sum^ \infty_{n = 0} a_ n x^ n\), where \(a_ n\) is the number of elements of \(G\) which are words of length \(n\) in \(S\). The Cayley graph

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of the pair \((G,S)\) is the directed graph

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whose vertices are the elements of \(G\) and there is an edge from a vertex \(g_ 1\) to a vertex \(g_ 2\) iff \(g_ 2 = g_ 1g\) for some \(g \in S \cup S^{-1}\).
In the paper the author investigates the walk in the Cayley graph of the “top” of the restricted wreath product

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\(G = K \wr H\), and gives a description of \(f_ G(x)\), when this graph is a tree. A large class of these growth series is shown to consist of irrational algebraic functions

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.

Reviewer: P.Lakatos (Debrecen)

MSC:

20F65	Geometric group theory
05C25	Graphs and abstract algebra (groups, rings, fields, etc.)
20E22	Extensions, wreath products, and other compositions of groups
20F05	Generators, relations, and presentations of groups

Keywords:

finite generating set; growth power series; Cayley graph; restricted wreath product

References:

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