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For positive integers u and v , let L u = [ 1 0 u 1 ] and R v = [ 1 v 0 1 ]. Let G u , v be the group generated by L u and R v. In a previous paper, the authors determined a characterization of matrices M = [ a b c d ] in G u , v when u , v ≥ 3 in terms of the short continued fraction representation of b / d. We extend this result to the case where u v ≥ 5. Additionally, we compute [ G u , v : G u , v ] where G u , v = { [ 1 + u v n 1 v n 2 u n 3 1 + u v n 4 ] ∈ SL 2 (Z) : (n 1 , n 2 , n 3 , n 4) ∈ Z 4 } for u , v ≥ 1 , extending a result of Chorna, Geller, and Shpilrain and solving the membership problem for G u , v with u , v ≥ 1. [ABSTRACT FROM AUTHOR]

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