In this course we present and discuss some elements of the geometric theory of 2-dimensional (bounded) homology from several points of view, making contact with low-dimensional geometry and topology, and with group dynamics. We start off at an elementary level, but we hope to present some original results which will also be of interest to the experts, and to outline a number of open problems which might make good thesis topics. There are few prerequisites for the class, since much of the material will be presented from scratch. Students should know about the topology of surfaces, some of the details of a first course in algebraic topology, and perhaps some hyperbolic geometry (though this will not be necessary if the student is prepared to take a couple of things on faith). Roughly, the plan of the course is:
The textbook for the course is my lecture notes scl.