Geometric group theory and arithmetic diameter
Abstract
Let X be a group with identity e, let A be an infinite set of generators for X, and let (X,d_A) be the metric space with the word metric d_A induced by A. If the diameter of the space is infinite, then for every positive integer h there are infinitely many elements x in X with d_A(e,x)=h. It is proved that if P is a nonempty finite set of prime numbers and A is the set of positive integers whose prime factors all belong to P, then the diameter of the metric space (\Z,d_A) is infinite. Let \lambda_A(h) denote the smallest positive integer x with d_A(e,x)=h. It is an open problem to compute \lambda_A(h) and estimate its growth rate.
 Publication:

arXiv eprints
 Pub Date:
 January 2011
 arXiv:
 arXiv:1101.0786
 Bibcode:
 2011arXiv1101.0786N
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Combinatorics;
 Mathematics  Group Theory;
 Mathematics  Metric Geometry;
 05A17;
 11B05;
 11B13;
 11P99;
 20F65
 EPrint:
 7 pages. Minor editorial changes and corrected typos