Finite quotients of triangle groups

Extending an explicit result from Bridson–Conder–Reid, this work provides an algorithm for distinguishing finite quotients between cocompact triangle groups Δ and lattices Γ of constant curvature symmetric 2-spaces. Much of our attention will be on when these lattices are Fuchsian groups. We prove that it will suffice to take a finite quotient that is Abelian, dihedral, a subgroup of PSL(n,F_q) (for an odd prime power q), or an Abelian extension of one of these 3 groups. For the latter case, we will require and develop an approach for creating group extensions upon a shared finite quotient of Δ and Γ which between them have differing degrees of smoothness. Furthermore, on the order of a finite quotient that distinguishes between Δ and Γ, we are able to establish an effective upperbound that is superexponential depending on the cone orders appearing in each group.