On congruence subgroups generated by two parabolic rational matrices

We study the freeness problem for multiplicative subgroups of SL₂(Q) . For $q = r/p$ in Q ∩ (0,4) , where p is prime and (r,p)=1 , we initiate the study of the algebraic structure of the group Δq generated by \[A = {pmatrix}1 & 0 \\ 1 & 1{pmatrix} and Q_q = {pmatrix}1 & q \\ 0 & 1{pmatrix}.\] We introduce the conjecture that Δr/p = Γ̄₁⁽ᵖ⁾(r) , the congruence subgroup of SL₂(Z[1/p]) consisting of all matrices with upper right entry congruent to 0 mod r and diagonal entries congruent to 1 mod r . We prove this conjecture when r ≤ 4 and for some cases when $r = 5$ . Furthermore, conditional on a strong form of Artin’s conjecture on primitive roots, we also prove the conjecture when r ∈ \ p-1, p+1, (p+1)/2 \ . In all these cases, this gives information about the algebraic structure of Δr/p : it is isomorphic to the fundamental group of a finite graph of virtually free groups, and has finite index J₂(r) in SL₂(Z[1/p]) , where J₂(r) denotes the Jordan totient function.