Geometric decompositions and asymptotic invariants of smooth manifolds

In 1982 Gromov introduced the minimal and simplicial volumes. These are related to infima of the asymptotic volume growth rate (also known as volume entropy) and of the topological entropy of the geodesic flow. After a brief description of these invariants I will give an overview of how certain geometric constructions (geometric decompositions in the sense of Thurston and JSJ decompositions of nonpositively curved manifolds) are used to show vanishing results for the minimal topological entropy, the minimal volume and for an invariant invented by Perelman which often equals the Yamabe invariant. I will explain that adding certain (nonessential) manifolds in a connected sum does not change the minimal volume entropy. This will then be used to present new examples of smooth four-manifolds whose homotopy type satisfies every restriction known so far to the existence of an Einstein metric, and yet despite this, they admit infinitely many smooth structures which do not admit Einstein metrics.