  We investigate the sub-Planck-scale structures associated with the SU(1,1)
group by establishing that the Planck scale on the hyperbolic plane can be
considered as the inverse of the Bargmann index $k$. Our discussion involves
SU(1,1) versions of Wigner functions, and the quantum-interference effect is
easily visualized through plots of these Wigner functions. Specifically, the
superpositions of four Perelomov SU(1,1) coherent states (compass state) yield
nearly isotropic sub-Planck structures in phase space scaling as $\frac1{k}$
compared with $\frac1{\sqrt{k}}$ scaling for individual SU(1,1) coherent states
and anisotropic quadratically improved scaling for superpositions of two
SU(1,1) coherent states (cat state). We show that displacement sensitivity
exhibits the same quadratic improvement to scaling.
