### Two problems in number theory related to geometric group theory

#### Melvyn Nathanson, CUNY Lehman

Abstract: The first problem arises from a question of R. E. Schwartz. Let a > 1 and let d_a be the word metric on the additive group of integers associated with the infinite generating set {1,a,a^2,a^3,a^4,...}. Schwartz asked if the metric spaces (Z, d_2) and (Z,d_3) are quasi-isometric. It is proved that for a > 1 and b > 1, the metrics d_a and d_b on Z are bi-Lipschitz equivalent if and only if a^m = b^n for some positive integers m and n.

The second problem arise from the "fundamental observation of geometric group theory," which asserts that if the action of a group of isometries on a proper geodesic metric space is properly discontinuous and co-compact, then the group is finitely generated, and the group action produces a set of generators. We ask the inverse question: What finite sets of generators arise geometrically from such a group action? We prove that in the case of the integers, every finite generating set (i.e. every finite symmetric set of relatively prime integers) can be constructed geometrically. The inverse problem for other groups, such as the additive group Z^2 of lattice points, seems much more difficult.