When viewed as a ( 2 : 2 ) (2:2) holomorphic correspondence on the Riemann sphere, the modular group P S L 2 ( Z ) PSL_2({\mathbb Z}) has a moduli space Q {\mathcal Q} of non-trivial deformations for which the limit set remains a topological circle. This space is analogous to a Bers slice of the deformation space of a Fuchsian group as a Kleinian group, but there are certain differences. A Bers slice contains a single quasiconformal conjugacy class of Kleinian groups: we show that for an open dense set of parameter values in Q {\mathcal Q} the correspondence belongs to a single quasi-conformal conjugacy class, but that at a countable set C {\mathcal C} of isolated parameter values it satisfies an additional critical relation. We classify these relations, propose ‘pleating coordinates’ for Q {\mathcal Q} , and investigate how the correspondence degenerates on the boundary of Q {\mathcal Q} . In particular, we show that there is a point on the boundary of Q {\mathcal Q} where the correspondence degenerates into a mating between P S L 2 ( Z ) PSL_2(\mathbb Z) and the quadratic polynomial z → z 2 + 1 / 4 z \to z^2+1/4 . A key ingredient in our analysis is a bijection between Q ∖ C {\mathcal Q}\setminus {\mathcal C} and an intermediate cover between the moduli space of the space of non-critical grand orbits of the correspondence, and its universal cover, the corresponding Teichmüller space.