Finite index rigidity of hyperbolic groups

We prove that the topological complexity of a finite index subgroup of a hyperbolic group is linear in its index. This follows from a more general result relating the size of the quotient of a free cocompact action of a hyperbolic group on a graph to the minimal number of cells in a simplicial classifying space for the group. As a corollary, we prove that any two isomorphic finite index subgroups of a non-elementary hyperbolic group have the same index.