On the structure of groups with polynomial growth. IV
Let \(G\) be a locally compact, compactly generated group. Let also \(\lambda\) be a Haar measure on \(G\) and let \(V\) be a compact neighbourhood of the identity \(e\), generating \(G\). The group \(G\) is said to be of \textit{polynomial growth}, if there exists \(d \in \mathbb{N}\) such that \(\lambda(V^n)=O(n^d)\) for \(n \in \mathbb{N}\). If \(G\) is \textit{almost nilpotent} (that is, \(G\) has a nilpotent subgroup \(N\) such that \(G/N\) is compact), then it has polynomial growth and Gromov characterized finitely generated discrete groups with polynomial growth precisely as the almost nilpotent ones (see [\textit{M. Gromov}, Publ. Math., Inst. Hautes Étud. Sci. 53, 53--78 (1981; Zbl 0474.20018)]). On the other hand, the converse is not true in non-discrete case (see [\textit{V. Losert}, J. Lond. Math. Soc., II. Ser. 63, No. 3, 640--654 (2001; Zbl 1010.22008)] for examples). In [J. Algebra 554, 1--40 (2020; Zbl 1440.22010)], the author generalized Gromov's result providing a structure for locally compact groups of polynomial growth and the aim of this paper is to give some applications on various growth functions and relations to the \(FC^{-}\)\!-series of \(G\). Recall that if \(G\) is a locally compact group, the \textit{upper \(FC^{-}\)\!-central series} of \(G\) is recursively defined as follows: \(B_0(G)=\{e\}\); \(B_1(G)\) is the closure of the set \(B(G)\) of all elements of \(G\) having a relatively compact orbit under conjugation; for any ordinal \(\alpha\), \(B_{\alpha+1}(G)/B_\alpha(G)\) is the closure of \(B(G/B_\alpha(G))\); and, if \(\lambda\) is a limit ordinal, then \(B_\lambda(G)\) is the closure of the union of the \(B_\alpha(G)\) with \(\alpha<\lambda\). The least ordinal \(\gamma\) such that \(B_{\gamma+1}(G)=B_\gamma(G)\) is the \(FC^-\)\!-\textit{class} of \(G\), and \(G\) is \(FC^-\)\!-\textit{hypercentral} if it coincides with \(B_\gamma(G)\). Among many results (which are far too complicated to be stated here) is proved that the \(FC^-\)\!-class of a compactly generated locally compact group is at most \(\omega_0\), and in case the group is \(FC^-\)\!-hypercentral, such a class is even finite (see Theorem 1.12). Many examples and applications are provided; and many related classes of groups are considered.
scientific article; zbMATH DE number 7354441
scientific article