Given a closed hyperbolic manifold M, are there lower bounds on the number of k-cells c_k(M) in a cell decomposition in terms of the geometry of the manifold? Gromov showed that if the manifold has injectivity radius at least 10^6 times (n log n), then there are at least n 1-cells, and conjectured that injectivity radius const times log n should be enough. In this talk I will describe a result providing a lower bound on the number of k-cells for each 0 < k < dim (M). The main input is a freedom theorem for ideals in group rings of hyperbolic groups, which also has other applications. Joint work with Thomas Delzant.