Groups of polynomial growth and their associated metric spaces
type: Item
instance of: scholarly article
title: Groups of polynomial growth and their associated metric spaces
zbMATH Open document ID: 0833.20046
author: Françoise Point
published in: Journal of Algebra
publication date: 1996-03-12T00:00:00Z
review text: 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 =…
reviewed by: Darryl McCullough
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20F18
Mathematics Subject Classification ID: 54D99
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 54H10
zbMATH DE Number: 783913
zbMATH Keywords: limited ultraproducts
zbMATH Keywords: finitely generated group
zbMATH Keywords: generating set
zbMATH Keywords: discrete metric space
zbMATH Keywords: action
zbMATH Keywords: isometry
zbMATH Keywords: growth function
zbMATH Keywords: polynomial growth
zbMATH Keywords: extensions of finite index
zbMATH Keywords: non principal ultrafilter
zbMATH Keywords: Minkowski dimension
zbMATH Keywords: finitely generated nilpotent group
zbMATH Keywords: descending central series
MaRDI profile type: Publication
full work available at URL: jabr.1995.1177
OpenAlex ID: W1988777076
DOI: 10.1006/JABR.1995.1177
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description: scientific article; zbMATH DE number 783913