### Abstract:

Let C(Γ) be the set of isomorphism classes of the finite groups that are quotients (homomorphic images) of Γ. We investigate the extent to which C(Γ) determines Γ when Γ is a group of geometric interest. If Γ1 is a lattice in PSL(2, R) and Γ2 is a lattice in any connected Lie group, then C(Γ1) = C(Γ2) implies that Γ1 ≅ Γ2. If F is a free group and Γ is a right-angled Artin group or a residually free group (with one extra condition), then C(F) = C(Γ) implies that F ≅ Γ. If Γ1 < PSL(2, C) and Γ2 < G are nonuniform arithmetic lattices, where G is a semisimple Lie group with trivial centre and no compact factors, then C(Γ1) = C(Γ2) implies that G ≅ PSL(2, C) and that Γ2 belongs to one of finitely many commensurability classes. These results are proved using the theory of profinite groups; we do not exhibit explicit finite quotients that distinguish among the groups in question. But in the special case of two non-isomorphic triangle groups, we give an explicit description of finite quotients that distinguish between them.