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

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 $n\le 1500$. Using the resulting series data we find that the growth rate of the group is bounded above by $\mathrm{2.62167...}$. This is very close to Guba's lower bound of $\frac{3+\sqrt{5}}{2}$ \cite{Guba2004}. Indeed, numerical analysis of the series data strongly suggests that the growth rate of the group is exactly $\frac{3+\sqrt{5}}{2}$.

27 pages, 9 figures, 3 tables

References in corpus (3)

• Bijections for Baxter Families and Related Objects
• Algorithms and Classification in Groups of Piecewise-Linear Homeomorphisms
• Some geodesic problems in groups

Cited by in corpus (2)

• Some geodesic problems in groups
• A computational approach to the Thompson group $F$