Solving the membership problem for certain subgroups of [formula omitted]

Abstract

For positive integers u and v, let Lu=[10u1] and Rv=[1v01]. Let Gu,v be the group generated by Lu and Rv. In a previous paper, the authors determined a characterization of matrices M=[abcd] in Gu,v when u,v≥3 in terms of the short continued fraction representation of b/d. We extend this result to the case where uv≥5. Additionally, we compute [Gu,v:Gu,v] where Gu,v={[1+uvn1vn2un31+uvn4]∈SL2(Z):(n1,n2,n3,n4)∈Z4} for u,v≥1, extending a result of Chorna, Geller, and Shpilrain and solving the membership problem for Gu,v with u,v≥1.