The modular group and words in its two generators*

Consider the full modular group PSL₂(?) with presentation 〈U,?S|U ³,?S ²〉. Motivated by our investigations on quasi-modular forms and the Minkowski question mark function (so that this paper may be considered as a necessary appendix), we are led to the following natural question. Some words in the alphabet {U, S} are equal to the unity; for example, USU ³ SU ² is such a word of length 8, and USU ³ SUSU ³ S ³ U is such a word of length 15. We consider the following integer sequence. For each n ∈ ?₀, let t(n) be the number of words in the alphabet {U, S} that equal the identity in the group. 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 ?(x) of degree 3. As an interesting generalization, we formulate the problem of describing all algebraic functions with a Fermat property.