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

Abstract

For positive integers u u and v v , let L u = [ 1 a m p ; 0 u a m p ; 1 ] L_u=\left [\begin {smallmatrix} 1 & 0 \\ u & 1 \end {smallmatrix}\right ] and R v = [ 1 a m p ; v 0 a m p ; 1 ] R_v=\left [\begin {smallmatrix} 1 & v \\ 0 & 1 \end {smallmatrix}\right ] . Let S u , v S_{u,v} be the monoid generated by L u L_u and R v R_v , and let G u , v G_{u,v} be the group generated by L u L_u and R v R_v . In this paper we expand on a characterization of matrices M = [ a a m p ; b c a m p ; d ] M=\left [\begin {smallmatrix}a & b \\c & d\end {smallmatrix}\right ] in S k , k S_{k,k} and G k , k G_{k,k} when k ≥ 2 k\geq 2 given by Esbelin and Gutan to S u , v S_{u,v} when u , v ≥ 2 u,v\geq 2 and G u , v G_{u,v} when u , v ≥ 3 u,v\geq 3 . We give a simple algorithmic way of determining if M M is in G u , v G_{u,v} using a recursive function and the short continued fraction representation of b / d b/d .