Description
Recent research has repeatedly led to connections between important rigidity questions and bounded cohomology. However, the latter has remained by and large intractable. This monograph introduces the functorial study of the continuous bounded cohomology for topological groups, with coefficients in Banach modules. The powerful techniques of this more general theory have successfully solved a number of the original problems in bounded cohomology. As applications, one obtains, in particular, rigidity results for actions on the circle, for representations on complex hyperbolic spaces and on Teichmller spaces. A special effort has been made to provide detailed proofs or references in quite some generality. Introduction Chapter I Banach modules, $L^/infty$ spaces 1 Banach modules 2 $L^/infty$ spaces 3 Integration Chapter II Relative injectivity and amenable actions 4 Relative injectivity 5 Amenability and amenable actions Chapter III Definition and characterization of continuous bounded cohomology 6 A naive definition 7 The functorial characterization 8 Functoriality 9 Continuous cohomology and the comparison map Chapter IV Cohomological techniques 10 General techniques 11 Double ergodicity 12 Hochschild-Serre spectral sequence Chapter V Towards applications 13 Interpretations of $(^2 /rm EH)rm cb)$ 14 General irreducible lattices Bibliography Index
