A symplectic manifold is called symplectically aspherical, if the cohomology class of its symplectic form vanishes on the image of the Hurewicz homomorphism. This class of manifolds has interesting topological properties and is a toolkit in symplectic topology. There are several natural problems in the area, for example the question about realization of groups as fundamental groups of symplectically aspherical manifolds. We prove several realization theorems for some important classes of groups, like polycyclic groups, some group extensions and amalgamated products. Also, we analyze the behaviour of symplectic asphericity under some classical symplectic constructions like symplectic surgery and symplectic fibrations.

  This is a joint work with Raul Ibanez, Jarek Kedra and Yuli Rudyak.