Groups of polynomial growth and their associated metric spaces
Associated to a finitely generated group \(G\) with generating set \(X\), there is a discrete metric space \((G, d_X)\) on which the action of each element of \(G\) by left multiplication is an isometry. The growth function \(G_X\) is defined by putting \(G_X(n)\) equal to the cardinality of the closed ball centered at the identity of \(G\) with radius \(n\). It is said that \(G\) has polynomial growth if there exist \(c\) and \(d\) such that \(G_X(n) \leq cn^d\) for all \(n\), in which case the growth degree is defined to be the smallest such \(d\). This is independent of the choice of \(X\) and is unchanged by passing to subgroups or extensions of finite index. J. Wolf proved that if \(G\) is nilpotent, then it has polynomial growth, and M. Gromov proved that if \(G\) has polynomial growth, then \(G\) is a finite extension of a nilpotent group. Gromov proved this by the remarkable device of showing that there is an increasing sequence of integers \(n_i\) such that the sequence of metric spaces \((G, d_X/n_i)\) converges (in an appropriate sense) to a metric space \(Y\) which is homogeneous, complete, geodesic, locally compact, and finite-dimensional. These ideas have undergone further development by later authors. As explained in the paper under review, Van den Dries and Wilkie introduced ``limited ultraproducts of metric spaces. Let \(U\) be a non principal ultrafilter on \(\omega\) and let \((n_i)\) be a sequence of natural numbers going to infinity. Let \(N = [n_i]_U\) belong to the ultrapower \(S^\omega/U\) [where \(S\) is the natural numbers]; we will call \(N\) a nonstandard natural number. Let \(G\) be any finitely generated group; we will denote by \(Y_N\) the limited ultraproduct of the \((G, d_X/n_i, e)\) quotiented out by the subgroup of the infinitesimals. It is a metric space which is complete, geodesic, and homogeneous. If moreover each \(n_i\) is \(i\)-regular, then the sequence \((G, d_X/n_i, e)\) has a subsequence converging to \(Y_N\) where \(N = [n_i]_U\) for some ultrafilter \(U\). If the finitely generated group \(G\) contains the free monoid \(M_2\) on two generators, then for no nonstandard natural number \(N\) is \(Y_N\) locally compact.'' The author summarizes the content of the paper under review as follows. ``Van den Dries and Wilkie asked whether local compactness of \(Y_N\) for some nonstandard natural number \(N\) implies that \(G\) is nilpotent-by- finite. We answer their question positively under the additional hypothesis that \(Y_N\) is not only locally compact, but has finite Minkowski dimension. This hypothesis comes from the fact that if \((n_i)\) is a sequence of natural numbers going to infinity such that each \(n_i\) is \(i\)-regular, then \(Y_N\) has finite Minkowski dimension.'' ``In the last part of the paper we prove that if \(G\) is a finitely generated nilpotent group, the space \(Y_N\) associated to \(G\) and the nonstandard natural number \(N\) is homeomorphic to [a Euclidean space] \(R^n\), where \(n\) is the sum of the ranks of the descending central series of \(G\)''.
scientific article; zbMATH DE number 783913