On congruence subgroups generated by two parabolic rational matrices

Abstract 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 q generated by = pmatrix1 & 0 \\ 1 & 1pmatrix and Qq = pmatrix1 & q \\ 0 & 1pmatrix. \ We introduce the conjecture that ₑ/₏ = ₁^ (p) (r), the congruence subgroup of SL₂ (Z1/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 ₑ/₏: it is isomorphic to the fundamental group of a finite graph of virtually free groups, and has finite index J₂ (r) in SL₂ (Z1/p), where J₂ (r) denotes the Jordan totient function.