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We study generalized two-field $\alpha$-attractor models whose rescaledscalar manifold is the triply-punctured sphere endowed with its completehyperbolic metric, whose underlying complex manifold is the modular curve$Y(2)$. Using an explicit embedding into the end compactification, we computesolutions of the cosmological evolution equations for a few globallywell-behaved scalar potentials, displaying particular trajectories withinflationary behavior as well as more general cosmological trajectories ofsurprising complexity. In such models, the orientation-preserving isometrygroup of the scalar manifold is isomorphic with the permutation group on threeelements, acting on $Y(2)$ as the group of anharmonic transformations. When thescalar potential is preserved by this action, $\alpha$-attractor models of thistype provide a geometric description of two-field `modular invariant$j$-models' in terms of gravity coupled to a non-linear sigma model withtopologically non-trivial target and with a finite (as opposed to discrete butinfinite) group of symmetries. The precise relation between the twoperspectives is provided by the elliptic modular function $\lambda$, which canbe viewed as a field redefinition that eliminates almost all of the countablyinfinite unphysical ambiguity present in the Poincar\'e half-plane descriptionof such models.

Generalized two-field $\alpha$-attractors from the hyperbolic triply-punctured sphere