  Cosmological alpha-attractors give a natural explanation for the spectral
index n_s of inflation as measured by Planck while predicting a range for the
tensor-to-scalar ratio r, consistent with all observations, to be measured more
precisely in future detection of gravity waves. Their embedding into
supergravity exploits the hyperbolic geometry of the Poincare disk or
half-plane. These geometries are isometric under Mobius transformations, which
include the shift symmetry of the inflaton field. We introduce a new Kahler
potential frame that explicitly preserves this symmetry, enabling the inflaton
to be light. Moreover, we include higher-order curvature deformations, which
can stabilize a direction orthogonal to the inflationary trajectory. We
illustrate this new framework by stabilizing the single superfield
alpha-attractors.
