  We investigate bipartite entanglement in spin-1/2 systems on a generic
lattice. For states that are an equal superposition of elements of a group $G$
of spin flips acting on the fully polarized state $\ket{0}^{\otimes n}$, we
find that the von Neumann entropy depends only on the boundary between the two
subsystems $A$ and $B$. These states are stabilized by the group $G$. A
physical realization of such states is given by the ground state manifold of
the Kitaev's model on a Riemann surface of genus $\mathfrak{g}$. For a square
lattice, we find that the entropy of entanglement is bounded from above and
below by functions linear in the perimeter of the subsystem $A$ and is equal to
the perimeter (up to an additive constant) when $A$ is convex. The entropy of
entanglement is shown to be related to the topological order of this model.
Finally, we find that some of the ground states are absolutely entangled, i.e.,
no partition has zero entanglement. We also provide several examples for the
square lattice.
