  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.
