The modular group and words in its two generators

TL;DR

This paper studies words in the modular group generated by U and S, focusing on counting words of a given length that equal the identity, and analyzes the algebraic nature of their generating function.

Contribution

It introduces a sequence counting identity words in the modular group and proves its generating function is algebraic of degree 3 over 1(1)(x).

Findings

Sequence A265434 identified and analyzed.

Generated function proven to be algebraic of degree 3.

Formulated problem on algebraic functions with Fermat property.

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.