The growth of finite subgroups of soluble groups. (Q1425527)

Two assertions are proved. (1) The growth of finite subgroups of finitely generated metabelian groups is either exponential or bounded. (2) Let \(g\colon\mathbb{N}\to\mathbb{N}\setminus\{0\}\) be an arbitrary nondecreasing function, where \(\mathbb{N}=\{0,1,2,\dots\}\) with the condition \(g(n)\leq\alpha^n\) for some \(\alpha>1\). Then there exists a 2-generated centrally metabelian group \(G^*\) the growth of finite subgroups of which equals \(\overline g\).