The geometry of spheres in free abelian groups

TL;DR

This paper investigates the geometric and probabilistic properties of word metrics on free abelian groups, establishing the existence of a limit measure on spheres and linking these to convex geometry for asymptotic analysis.

Contribution

It introduces new tools to analyze dependence on generating sets and proves the convergence of counting measures to a limit measure on the shape of spheres in free abelian groups.

Findings

Counting measure on spheres converges to a limit measure.

Asymptotic formulas for spherical growth functions.

Probabilistic questions reduce to convex geometry problems.

Abstract

We study word metrics on Z^d by developing tools that are fine enough to measure dependence on the generating set. We obtain counting and distribution results for the words of length n. With this, we show that counting measure on spheres always converges to a limit measure on a limit shape (strongly, in an appropriate sense). The existence of a limit measure is quite strong-even virtually abelian groups need not satisfy these kinds of asymptotic formulas. Using the limit measure, we can reduce probabilistic questions about word metrics to problems in convex geometry of Euclidean space. As an application, we give asymptotics for the spherical growth function with respect to any generating set, as well as statistics for other "size-like" functions.