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,dA) be the metric space with the word metric dA 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 dA(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 (,dA) is infinite. Let λA(h) denote the smallest positive integer x with dA(e,x)=h. It is an open problem to compute λA(h) and estimate its growth rate.
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