Surfaces and bounded cohomology
We introduce the notion of causal representation of a surface group and relate it to that of maximal representation and of tight homomorphism. When the target is SL(2,R) we show that these are hyperbolizations. In the process we define and study the bounded fundamental class of a compact surface (with or without boundary) and establish a result characterizing it among all bounded classes. We relate this to the winding number of Chillingsworth and to work of Calegari on stable commutator length.