I am starting my Master's degree, and I have to chose between these two courses that I have not taken during my Bachelor's (I can only chose one!). I am interested in group theory and in particular geometric group theory. On the one hand, I'd like to do differential geometry to have a stronger basis for studying Lie groups. On the other hand, I know that the study of analytical properties of groups, such as everything around the notion of amenability, relies on some results of functional analysis.
If any of you works in (geometric) group theory, what do you think is most important for a mathematician working in that field to master, and what is easier to learn by oneself? Thanks in advance.
Hell, I'm interested mainly in semigroups and automata and I did both FA and Differential geometry. Took FA even further and took a course on the theory of topological vector spaces.
What I'm trying to say is don't be too worried about specialising in something very specific. I'm on my 2nd year in master's programme and definitely plan on doing PhD on top of this. So there's plenty of time. Widen your horizon on the basics of several interesting fields :)