  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.
