I'm interested in the structure of homeo- and diffeomorphism groups of oriented surfaces, especially in hyperbolic case. For example, does the homeomorphism group retracts on the diffeomorphism group (fixing smooth structure) or the isometry group (fixing a Riemann metric with constant curvature)?
So could you give me references on the articles, or better, books about these topics? I'd also like to read the motivations and standard examples of topologies on these groups, isotopies of different structures and other technical things.