Bi-Lipschitz equivalent metrics on groups, and a problem in additive number theory

TL;DR

This paper investigates when different word length metrics on groups are bi-Lipschitz equivalent, linking this property to the existence of positive integer solutions to exponential equations in additive number theory.

Contribution

It establishes a precise criterion for bi-Lipschitz equivalence of metrics derived from different generating sets in groups, connecting geometric group theory with additive number theory.

Findings

Metrics are bi-Lipschitz equivalent iff a^m = b^n for some positive integers m, n

Provides a new perspective on the structure of groups via metric equivalence

Links group metrics to exponential Diophantine equations

Abstract

There is a standard "word length" metric canonically associated to any set of generators for a group. In particular, for any integers a and b greater than 1, the additive group of integers has generating sets {a^i}_{i=0}^{\infty} and {b^j}_{j=0}^{\infty} with associated metrics d_A and d_B, respectively. It is proved that these metrics are bi-Lipschitz equivalent if and only if there exist positive integers m and n such that a^m = b^n.