Monodromy and birational geometry of O'Grady's sixfolds

We prove that the bimeromorphic class of a hyperk"ahler manifold deformation equivalent to O’Grady’s six dimensional one is determined by the Hodge structure of its Beauville-Bogomolov lattice by showing that the monodromy group is maximal. As applications, we give the structure for the K"ahler and the birational K"ahler cones in this deformation class and we prove that the existence of a square zero divisor implies the existence a rational lagrangian fibration with fixed fibre types.