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We consider four-dimensional gravity coupled to a non-linear sigma modelwhose scalar manifold is a non-compact geometrically finite surface $\Sigma$endowed with a Riemannian metric of constant negative curvature. When thespace-time is an FLRW universe, such theories produce a very widegeneralization of two-field $\alpha$-attractor models, being parameterized by apositive constant $\alpha$, by the choice of a finitely-generated surface group$\Gamma\subset \mathrm{PSL}(2,\mathbb{R})$ (which is isomorphic with thefundamental group of $\Sigma$) and by the choice of a scalar potential definedon $\Sigma$. The traditional two-field $\alpha$-attractor models arise when$\Gamma$ is the trivial group, in which case $\Sigma$ is the Poincar\'e disk.We give a general prescription for the study of such models throughuniformization in the so-called "non-elementary" case and discuss some of theirqualitative features in the gradient flow approximation, which we relate toMorse theory. We also discuss some aspects of the SRST approximation in thesemodels, showing that it is generally not well-suited for studying dynamics nearcusp ends. When $\Sigma$ is non-compact and the scalar potential is"well-behaved" at the ends, we show that, in the {\em naive} local one-fieldtruncation, our generalized models have the same universal behavior as ordinaryone-field $\alpha$-attractors if inflation happens near any of the ends of$\Sigma$ where the extended potential has a local maximum, for trajectorieswhich are well approximated by non-canonically parameterized geodesics near theends, we also discuss spiral trajectories near the ends.

Generalized two-field $\alpha$-attractor models from geometrically finite hyperbolic surfaces