Fine asymptotic geometry in the Heisenberg group

TL;DR

This paper investigates the large-scale geometric structure of word metrics in the integer Heisenberg group, revealing how properties depend on the choice of generators and connecting to asymptotic geometric limits.

Contribution

It characterizes the asymptotic geometry of word metrics in H(Z) and quantifies the probability of geodesic spellings being sublinearly close, depending on generators.

Findings

Probability varies between 0 and 1 depending on the group type

For standard generators in H(Z), probability is 19/31

Limits of word metrics relate to Carnot-Caratheodory Finsler metrics

Abstract

For every finite generating set on the integer Heisenberg group H(Z), Pansu showed that the word metric has the large-scale structure of a Carnot-Caratheodory Finsler metric on the real Heisenberg group H(R). We study the properties of those limit metrics and obtain results about the geometry of word metrics that reflect the dependence on generators. For example we will study the probability that a group element has all of its geodesic spellings sublinearly close together, relative to the size of the element. In free abelian groups of rank at least two, that probability is 0; in infinite hyperbolic groups, the probability is 1. In H(Z) it is a rational number strictly between 0 and 1 that depends on the generating set; with respect to the standard generators, the probability is precisely 19/31.