The modular group and words in its two generators*

Abstract

Consider the full modular group $\sf{PSL}_{2}(\mathbb{Z})$ with presentation $\langle U,S|U^3,S^2\rangle$. Motivated by our investigations on quasi-modular forms and the Minkowski question mark function (so that this paper might be considered as a necessary appendix), we are lead to the following natural question. Some words in the alphabet $\{U,S\}$ are equal to the unity; for example, $USU^3SU^2$ is such a word of length $8$, and $USU^3SUSU^3S^3U$ is such a word of length $15$. Given $n\in\mathbb{N}_{0}$. Find the number of words of length $n$ which are equal to the unity. This is the new entry A265434 into the Online Encyclopedia of Integer Sequences. We investigate the generating function of this sequence and prove that it is an algebraic function over $\mathbb{Q}(x)$ of degree $3$. As an aside, we formulate the problem of describing all algebraic functions with a Fermat property.