I'm a master's student in the Turin University. At the end of my studies, I have to write a master thesis. My main interest is geometric group theory, but it is not a research area of the Turin's mathematical department.
My professors in algebra and geometry are principally interested in algebraic geometry, commutative algebra or in Lie groups. My analysis professors are interested mainly in PDE and differential forms. So neither the theory of non-positive curved spaces, nor measure group theory seems to be a feasible alternative.
Since I want to do a PhD – not in Turin – I don't want to do a thesis too distant from my main interest. What I want to know is then if you know some topics of geometric group theory (or other similar theories) that are sufficiently connected with algebraic geometry (for example, with riemannian surfaces) or with Lie Groups and can be explored by a master student.
I haven't had yet time to explore deeply the main book on the subject – de la Harpe, Geoghegan, Bridson-Haefliger, Bowditch, Serre, Farb(on mapping class groups). Anyway, I suppose the theory presented in the Farb's book is the one that has more connections with the topics I listed before, or I'm wrong?