IUMJ

Title: Fine asymptotic geometry in the Heisenberg group

Authors: Moon Duchin and Christoper Mooney

Issue: Volume 63 (2014), Issue 3, 885-916

Abstract:

For every finite generating set on the integer Heisenberg group $H(\mathbb{Z})$, we know from a fundamental result of Pansu on nilpotent groups that the word metric has the large-scale structure of a Carnot-Carath\'eodory Finsler metric on the real Heisenberg group $H(\mathbb{R})$. We study the properties of those limit metrics and obtain results about the geometry of word metrics that reflect the dependence on generators, including asymptotic density results for geometric properties. For example, stability of geodesics and distortion of subgroups can be made statistical. This contributes to a small literature on asymptotic density results that depend nontrivially on generators for nonfree groups. Our methods also allow us to a pursue a "geometry of numbers" for nilpotent groups.