×

On rates of growth of p-groups and torsion-free groups. (Russian) Zbl 0568.20033

J. Milnor [Am. Math. Mon. 75, 685-686 (1968)] posed the problem whether there exist groups of intermediate growth, i.e. finitely generated groups with growth function not equivalent neither to any power function \(n^ k\) nor to the exponential function \(2^ n\). Recently the author [Izv. Akad. Nauk SSSR, Ser. Mat. 48, No.5, 939-983 (1984)] solved this problem by constructing a 2-group of intermediate growth. In the present article, the author constructs new examples of groups of intermediate growth in the class of p-groups for any prime p (Theorems 3, 4) and in the class of torsion-free groups (Theorem 5). The required p- groups are defined as those of interchanges of the \(p^ ith\) parts of the segment [0,1] or, equivalently, as groups of automorphisms of a homogeneous tree with branching multiplicity p, and the required torsion- free groups are constructed by an original combinatorical method whose idea is suggested by the algorithm solving the word problem in 2-groups (see ibid.). It is indicated in the article that a similar construction of p-groups of intermediate growth is obtained independently by I. G. Lysyonok.
Reviewer: Yu.I.Merzlyakov

MSC:

20F05 Generators, relations, and presentations of groups
20F50 Periodic groups; locally finite groups
20E34 General structure theorems for groups
PDFBibTeX XMLCite
Full Text: EuDML