Subgroups of 𝑆𝐿₂(ℤ) characterized by certain continued fraction representations

Abstract

For positive integers u and v, let Lu = 1 0 u 1 and Rv = 1 v 0 1 . Let Su,v be the monoid generated by Lu and Rv, and Gu,v be the group generated by Lu and Rv. In this paper we expand on a characterization of matrices M = a b c d in S k,k and G k,k when k ≥ 2 given by Esbelin and Gutan to Su,v when u, v ≥ 2 and Gu,v when u, v ≥ 3. We give a simple algorithmic way of determining if M is in Gu,v using a recursive function and the short continued fraction representation of b/d.

Solving the membership problem for certain subgroups of $SL_2(\mathbb{Z})$

“…By defining f u,v subject to u and v, we gain the necessary foothold missing from before when dealing with the u = 2 or v = 2 case in the membership problem. We now use our work in [2] as a template to extend Theorem 2.…”

“…Note that Lemma 2 represents a correction of Lemma 3.5 in [2] where the (−1) β term was mistakenly given as −1. The correction does not invalidate the alternate proof of Sanov's result given in [2], however the previous incorrect version can give product representations for matrices in G 2,2 whose exponents are off by a sign.…”

“…These definitions may initially seem unmotivated without some additional background. As this paper is designed to extend a previous result, we suggest that the reader consult [2] for a more thorough treatment motivating the above definitions. We are now able to state the result from [2] that we intend to extend.…”

“…The aim of this article is to complete our work from [2] on the membership problem for certain subgroups of SL 2 (Z). In order to state previous results and our extension, we must first introduce several definitions.…”

“…In [2], we showed that Theorem 1 could be modified and written in terms of the short continued fraction representations of either c/a or b/d, when u, v ≥ 3. In particular, we developed a simple algorithm that, when applied to the short continued fraction representation of b/d, determines whether or not the sought after continued fraction expansion in Theorem 1 exists.…”