Generalized two-field $\alpha$-attractors from the hyperbolic
  triply-punctured sphere
  We study generalized two-field $\alpha$-attractor models whose rescaled
scalar manifold is the triply-punctured sphere endowed with its complete
hyperbolic metric, whose underlying complex manifold is the modular curve
$Y(2)$. Using an explicit embedding into the end compactification, we compute
solutions of the cosmological evolution equations for a few globally
well-behaved scalar potentials, displaying particular trajectories with
inflationary behavior as well as more general cosmological trajectories of
surprising complexity. In such models, the orientation-preserving isometry
group of the scalar manifold is isomorphic with the permutation group on three
elements, acting on $Y(2)$ as the group of anharmonic transformations. When the
scalar potential is preserved by this action, $\alpha$-attractor models of this
type provide a geometric description of two-field `modular invariant
$j$-models' in terms of gravity coupled to a non-linear sigma model with
topologically non-trivial target and with a finite (as opposed to discrete but
infinite) group of symmetries. The precise relation between the two
perspectives is provided by the elliptic modular function $\lambda$, which can
be viewed as a field redefinition that eliminates almost all of the countably
infinite unphysical ambiguity present in the Poincar\'e half-plane description
of such models.
