Lithuanian Mathematical Journal

, Volume 57, Issue 1, pp 1–12 | Cite as

The modular group and words in its two generators*

  • Giedrius Alkauskas


Consider the full modular group PSL2(ℤ) 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 ∈ ℕ0, 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 ℚ(x) of degree 3. As an interesting generalization, we formulate the problem of describing all algebraic functions with a Fermat property.


the modular group combinatorial group theory free group unity generating function algebraic function cogrowth rate return generating function word problem pushdown automaton 


05A05 05A15 20XX 14H05 


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© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Faculty of Mathematics and InformaticsVilnius UniversityLT-03225Lithuania

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