Group Width

TL;DR

This paper introduces a new width function for countable groups, demonstrating that for the free abelian group Z^k, the width equals k-1, linking geometric and algebraic complexity measures.

Contribution

It defines a novel width function on groups and computes its value for Z^k, connecting group theory with geometric complexity concepts.

Findings

w(Z^k) = k-1 for free abelian groups

Establishes a link between width and algebraic structure

Provides a new perspective on group complexity measures

Abstract

There are many "minimax" complexity functions in mathematics: width of a tree or a link, Heegaard genus of a 3-manifold, the Cheeger constant of a Riemannian manifold. We define such a function w, "width", on countable (or finite) groups and show w(Z^k) = k-1.