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

We consider four-dimensional gravity coupled to a non-linear sigma model whose scalar manifold is a non-compact geometrically finite surface $\Sigma$ endowed with a Riemannian metric of constant negative curvature. When the space-time is an FLRW universe, such theories produce a very wide generalization of two-field $\alpha$-attractor models, being parameterized by a positive constant $\alpha$, by the choice of a finitely-generated surface group $\Gamma\subset \mathrm{PSL}(2,\mathbb{R})$ (which is isomorphic with the fundamental group of $\Sigma$) and by the choice of a scalar potential defined on $\Sigma$. 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 through uniformization in the so-called "non-elementary" case and discuss some of their qualitative features in the gradient flow approximation, which we relate to Morse theory. We also discuss some aspects of the SRST approximation in these models, showing that it is generally not well-suited for studying dynamics near cusp 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-field truncation, our generalized models have the same universal behavior as ordinary one-field $\alpha$-attractors if inflation happens near any of the ends of $\Sigma$ where the extended potential has a local maximum, for trajectories which are well approximated by non-canonically parameterized geodesics near the ends, we also discuss spiral trajectories near the ends.