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Bounded cohomology with a view towards hyperbolic geometry



Bounded cohomology is a relatively young theory introduced by Gromov in 1982. It differs from ordinary singular or group cohomology in that instead of considering arbitrary cochains, one restricts to cochains which have finite supremum norm. As a result, bounded cohomology behaves very differently from ordinary cohomology. A fundamental difference is that the bounded cohomology of a (reasonable) topological space is isomorphic to the bounded cohomology of its fundamental group. As a consequence, topological problems translate into questions about the fundamental group, such as understanding its space of representations.

In this subject, one topological invariant of particular interest for its connections with geometry is the simplicial volume of closed oriented manifolds. For example, for Riemannian manifolds, it is proportional to the volume by a constant depending only on the universal covering, a phenomenon which is familiar from Hirzebruch's proportionality principle. In particular, it follows at once that the volume of Riemannian manifolds is a topological invariant.

In general, simplicial volume computations, and a fortiori the determination of bounded cohomology groups, are very difficult. However, one instance where much can be said is for hyperbolic manifolds. Therefore, in this minicourse, after giving the first definitions and examples, I will concentrate on powerful applications of the theory to hyperbolic geometry. In particular I will discuss Milnor-Wood inequalities and nonexistence of affine structures on hyperbolic surfaces and products of hyperbolic surfaces, the proportionality principle for the simplicial volume for hyperbolic manifolds, and a simple geometric proof due to Gromov of Mostow's Rigidity Theorem for hyperbolic manifolds of dimension greater or equal than 3.