Abstract
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.
Similar content being viewed by others
References
G. Alkauskas, The Minkowski ?(x) function, a class of singular measures, quasi-modular and mean-modular forms, http://arxiv.org/abs/1209.4588.
L. Bartholdi, Counting paths in graphs, Enseign. Math. (2), 45(1–2):83–131, 2005.
S.R. Finch, Mathematical Constants, Encycl. Math. Appl., Vol. 94, Cambridge University Press, Cambridge, 2003.
J.E. Hopcroft and J.D. Ullman, Introduction to Automata Theory, Languages, and Computation, Addison-Wesley Series in Computer Science, Addison-Wesley, Reading, MA, 1979.
D.G. Kouksov, On rationality of the cogrowth series, Proc. Am. Math. Soc., 126(10):2845–2847, 1998.
D. Kuksov, Cogrowth series of free products of finite and free groups, Glasgow Math. J., 41(1):19–31, 1999.
F. Lehner, On the computation of spectra in free probability, J. Funct. Anal., 183(2):451–471, 2001.
J.C. McLaughlin, RandomWalks and Convolution Operators on Free Products, PhD dissertation, New York University, 1986.
G. Quenell, Combinatorics of free product graphs, in R. Brooks, C. Gordon, and P. Perry (Eds.), Geometry of the Spectrum. 1993 Joint Summer Research Conference on Spectral Geometry, July 17–23, University of Washington, Seattle, Contemp. Math., Vol. 173, AMS, Providence, RI, 1994, pp. 257–281.
D. Voiculescu, Addition of certain noncommuting random variables, J. Funct. Anal., 66(3):323–346, 1986.
W. Woess, Nearest neighbour random walks on free products of discrete groups, Boll. Unione Mat. Ital., VI Ser., B, 5(3):961–982, 1986.
W. Woess, Random walks on infinite graphs and groups, Camb. TractsMath., Vol. 138, Cambridge University Press, Cambridge, 2000.
The Online Encyclopedia of Integer Sequences, Sequence A265434, http://oeis.org/.
Author information
Authors and Affiliations
Corresponding author
Additional information
* The research of the author was supported by the Research Council of Lithuania grant No. MIP-072/2015.
Rights and permissions
About this article
Cite this article
Alkauskas, G. The modular group and words in its two generators* . Lith Math J 57, 1–12 (2017). https://doi.org/10.1007/s10986-017-9339-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10986-017-9339-2