Metric properties of higher-dimensional Thompson's groups

TL;DR

This paper investigates the metric properties of higher-dimensional Thompson's groups, providing bounds on word length and demonstrating the optimality and generic achievement of these bounds.

Contribution

It offers new descriptions of elements via tree-pair diagrams and establishes bounds on word length, advancing understanding of the metric structure of nV groups.

Findings

Upper and lower bounds for word length are established.

Bounds are shown to be optimal and generically achieved.

Elements realizing the lower bounds are constructed.

Abstract

Higher-dimensional Thompson's groups nV are finitely presented groups described by Brin which generalize dyadic self-maps of the unit interval to dyadic self-maps of n-dimensional unit cubes. We describe some of the metric properties of higher-dimensional Thompson's groups. We give descriptions of elements based upon tree-pair diagrams and give upper and lower bounds for word length in terms of the size of the diagrams. Though these upper and lower bounds are somewhat separated, we show that there are elements realizing the lower bounds and that the fraction of elements which are close to the upper bound converges to 1, showing that the bounds are optimal and that the upper bound is generically achieved.