Free ideals and cellular waist inequalities in hyperbolic geometry

I will describe a result showing for a closed hyperbolic manifold M and a (sufficiently nice) map f:M -> R^m, that there is a fiber whose topological complexity (measured by the number of cells in a cell structure) is bounded below in terms of the injectivity radius of M. It is based on freeness of ideals in group rings of hyperbolic groups, and addresses a recent question of Gromov. Joint work with Thomas Delzant.