The growth-rate conjecture for Thompson's group F

Let $F$ be Thompson's group with its standard generating set, and let $f(n)$ denote the number of elements of word length at most $n$. Define the growth rate by

$$\gamma=\lim_{n\to\infty}f(n)^{1/n}.$$

Growth-rate conjecture. The growth rate is

$$\gamma=\frac{3+\sqrt{5}}{2}.$$

The preceding result gives the rigorous bounds $\frac{3+\sqrt{5}}{2}\leq\gamma\leq 2.62167\ldots$, based on computation through $n=1500$ and a lower bound due to Guba. The conjecture asserts that the lower bound is exact.