Geometric group theory and arithmetic diameter
Number Theory
2014-01-03 v2 Combinatorics
Group Theory
Metric Geometry
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.
Cite
@article{arxiv.1101.0786,
title = {Geometric group theory and arithmetic diameter},
author = {Melvyn B. Nathanson},
journal= {arXiv preprint arXiv:1101.0786},
year = {2014}
}
Comments
7 pages. Minor editorial changes and corrected typos