Strict hyperbolization and special cubulation

Gromov introduced some procedures to turn a given polyhedron into a new one endowed with a piecewise Euclidean metric of non-positive curvature, while preserving some of its original topological features. Charney and Davis have proposed a refinement of Gromov's construction in which the new space carries a strictly negatively curved metric, and thus has hyperbolic fundamental group. After reviewing these constructions, we will discuss the problem of cubulating these groups, and some applications. This is joint work with J. Lafont.