Problem of intermediate relative growth of subgroups in solvable and linear groups (Q2783059)

For a group \(G\) with finite generating set \(X=\{x_1,\dots,x_s\}\) the length \(|g|\) of an element \(g\in G\) with respect to \(X\) is defined to be the length of the shortest word in the alphabet \(X\cup X^{-1}\) representing \(g\). The growth function of \(G\) is the function \(\beta_G\colon\mathbb{N}\cup\{0\}\to\mathbb{N}\) (\(\mathbb{N}\) is the set of positive integers) defined as: \(\beta_G(n)=\#\{g\in G\mid|g|\leq n\}\). \textit{R. I. Grigorchuk} [Adv. Probab. relat. Top. 6, 285-325 (1980; Zbl 0475.60007)] introduced the notion of relative growth: for an arbitrary (not necessarily finitely generated) subgroup \(H\) of \(G\) we can define the relative growth function as following: \(\gamma_H(n)=\#\{h\in H\mid|h|\leq n\}\). \textit{R. Grigorchuk} and \textit{P. de la Harpe} posed [in J. Dyn. Control Syst. 3, No. 1, 51-89 (1997; Zbl 0949.20033)] the following problem: Is it true that the relative growth of any subgroup of a solvable or linear group is either polynomial or exponential? The current paper gives a negative answer to this question for both group classes mentioned. A few particular cases when the answer to this problem is positive are considered.NEWLINENEWLINENEWLINEGrigorchuk [loc. cit.] also considered the following analog of the first problem: Is it true that the relative growth of any semigroup of a solvable or linear group is either polynomial or exponential? Here a negative answer is given for both cases, and a few particular cases when a positive answer is possible are considered.NEWLINENEWLINEFor the entire collection see [Zbl 0981.00006].