back to SummerSchool web page
## Bounded cohomology with a view towards hyperbolic geometry

###
Dr. Michelle BUCHER-KARLSSON

#### Abstract:

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.