  We construct families of cell complexes that generalize expander graphs.
These families are called non-$k$-hyperfinite, generalizing the idea of a
non-hyperfinite (NH) family of graphs. Roughly speaking, such a complex has the
property that one cannot remove a small fraction of points and be left with an
object that looks $k-1$-dimensional at large scales. We then consider certain
quantum systems on these complexes. A future goal is to construct a family of
Hamiltonians such that every low energy state has topological order as part of
an attempt to prove the quantum PCP conjecture. This goal is approached by
constructing a toric code Hamiltonian with the property that every low energy
state without vertex defects has topological order, a property that would not
hold for any local system in any lattice $Z^d$ or indeed on any 1-hyperfinite
complex. Further, such NH complexes find application in quantum coding theory.
The hypergraph product codes[1] of Tillich and Z\'{e}mor are generalized using
NH complexes.
