A group of intermediate growth acting by homomorphisms on the real line
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 \(\text{Aut}(\mathbb{R})\).
scientific article; zbMATH DE number 567872