Counting elements and geodesics in Thompson's group $F$

TL;DR

This paper introduces two algorithms for counting elements and geodesics in Thompson's group F, providing insights into its growth rate and demonstrating near-optimal bounds through computational methods.

Contribution

It presents one exponential-time algorithm with geodesic counting and a polynomial-time algorithm for large radii, advancing computational understanding of Thompson's group F.

Findings

Growth rate bounded above by approximately 2.62167

Numerical evidence suggests the exact growth rate is (3+√5)/2

Algorithms enable efficient counting of elements and geodesics in the group

Abstract

We present two quite different algorithms to compute the number of elements in the sphere of radius $n$ of Thompson's group $F$ with standard generating set. The first of these requires exponential time and polynomial space, but additionally computes the number of geodesics and is generalisable to many other groups. The second algorithm requires polynomial time and space and allows us to compute the size of the spheres of radius $n$ with $n \leq 1500$. Using the resulting series data we find that the growth rate of the group is bounded above by $2.62167...$. This is very close to Guba's lower bound of $\tfrac{3+\sqrt{5}}{2}$ \cite{Guba2004}. Indeed, numerical analysis of the series data strongly suggests that the growth rate of the group is exactly $\tfrac{3+\sqrt{5}}{2}$.