The modular group and words in its two generators*

Abstract

Consider the full modular group PSL 2 (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 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, U SU 3 SU 2 is such a word of length 8, and U SU 3 SU SU 3 S 3 U is such a word of length 15. We consider the following integer sequence. For each n ∈ N 0 , let t(n) be the number of words in 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.

Cogrowth Series for Free Products of Finite Groups

“…Computing the cogrowth of free products of free groups has been done in a number of cases [1], [12], [10]. We note that Kuksov's work is the most general, but as we have already remarked, he computes cogrowth using an altered definition.…”

“…These two counting sequences appear the Online Encyclopedia of Integer Sequences [18] as sequences A183135 and A265434. Alkauskas [1] worked out Theorem 1.2(c) in the case when m = 2 and n = 3, refining it to actually get the cubic equation for the cogrowth series (this is of special significance because this corresponds to the group PSL 2 (Z)). Theorem 1.3 was noticed as a curiosity coming from computations done while proving Theorem 1.2 and working out other examples.…”

On the complexity of the cogrowth sequence

“…We give an explicit algebraic system satisfied by the generating function. Alkauskas [1] does a computation for PSL 2 (Z), which is a free product of a cyclic group of order 2 and a cyclic group of order 3, using a non-symmetric generating set of size 2.…”

“…Research of the second-named author supported by NSERC Grant RGPIN-2017-04157. 1 mentioned, groups with context-sensitive word problem have not been classified, but an important result in this direction is that finitely generated subgroups of automatic groups have context-sensitive word problem [32]. These classes provide a taxonomy of groups in terms of the complexity of their word problem.…”

Evolution of ambiguous numbers under the actions of a Bianchi group

“…The action of Bianchi group 3 has played a very important role in the classification of the orbits of Q(i, √ 3). For detailed results and discussion related to Bianchi groups readers referred to [1][2][3]5,10,15,[25][26][27][28][29][30][31][32][33]. It is obvious to see the application of group theory to mechanics and physics to construct models, drive differential equations and investigate their structures [34][35][36].…”