  A hyperbolic conjugacy class in the modular group PSL(2,Z) corresponds to a
closed geodesic in the modular orbifold. Some of these geodesics virtually
bound immersed surfaces, and some do not; the distinction is related to the
polyhedral structure in the unit ball of the stable commutator length norm. We
prove the following stability theorem: for every hyperbolic element of the
modular group, the product of this element with a sufficiently large power of a
parabolic element is represented by a geodesic that virtually bounds an
immersed surface.
