Groups of Polynomial Growth and Their Associated Metric Spaces

Abstract

Let YN be the space associated with a finitely generated group G and a nonstandard natural number N. We prove the following two results. If YN is locally compact and has finite Minkowski dimension, then G is nilpotent-by-finite. If G is nilpotent, then YN is homeomorphic to Rn, where n is the sum of the ranks of the descending central series of G.