A group of intermediate growth acting by homomorphisms on the real line
type: Item
instance of: scholarly article
title: A group of intermediate growth acting by homomorphisms on the real line
zbMATH Open document ID: 0806.20027
DOI: 10.1007/BF01208318
author: V. Pereyra
published in: Mathematical Notes
publication date: 1994-07-13T00:00:00Z
review text: If \(G\) is a group the growth function relative to a finite system of generators \(A\) is the following function of natural numbers: \(\gamma(n) = \#\{g \in G: \partial(g) \leq n\}\), where \(\partial(g)\) is the length of \(g\) relative to \(A\) and \(\#(E)\) is the cardinality of the set \(E\). The order of growth of this function is independent of the choice of the generating system and does not exceed that of the exponential function \([2\#(A)]^ n\). A group \(G\) is said to have intermediate growth if \(\gamma(n)\) increases more rapidly than any power function \(n^ d\) but more slowly than any exponential function \(a^ n\), \(a > 1\). In 1968 \textit{J. Milnor} posed [in Problem 5603, Am. Math. Mon.] the problem of the existence of a subgroup of intermediate growth of \(\text{Aut}(\mathbb{R})\) and in 1984 the first author answered this question positively, giving an example of a torsion-free group. This paper continues the study of this group proving that this group is residually solvable and almost nilpotent and further, that it is embeddable into \(\text{Aut}(\mathbb{R})\).
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 06F15
Mathematics Subject Classification ID: 20F19
Mathematics Subject Classification ID: 20F60
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 20E26
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 567872
zbMATH Keywords: growth function
zbMATH Keywords: finite system of generators
zbMATH Keywords: subgroup of intermediate growth
zbMATH Keywords: torsion-free group
zbMATH Keywords: residually solvalbe
zbMATH Keywords: almost nilpotent
reviewed by: Grigore Călugăreanu
MaRDI profile type: Publication
cites work: Groups of piecewise linear homeomorphisms of the real line
cites work: Q3684302
cites work: The lattice-ordered group of automorphisms of an ordered set
cites work: Leaves whose growth is neither exponential nor polynomial
cites work: Relatively convex subgroups of orderable groups
description: scientific article; zbMATH DE number 567872