# Geometric group theory and arithmetic diameter

@article{Nathanson2011GeometricGT,
title={Geometric group theory and arithmetic diameter},
author={Melvyn B. Nathanson},
journal={arXiv: Number Theory},
year={2011}
}
• M. Nathanson
• Published 4 January 2011
• Mathematics
• arXiv: Number Theory
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… Expand
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