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

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

2009-02-18

arxiv.org/abs/0902.3254v1

Portug. Math. 68 (2011), 191--203

Mathematics

Metric Geometry

8 pages

Scientific paper

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.

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