I have just started taking a look at some introductory papers about Shimura varieties, after a friend of mine suggested me to do so. They seem to be a sort of very interesting and many-sided topic, but the approach I am following is the algebraic geometric one, as he studies algebraic geometry. Since I am not an algebraic geometer, I was wondering whether there exist some more topological aspects of those complex objects, perhaps even related with hyperbolic geometry, topology or so. I know that Shimura varieties are mainly used by number theorists and algebraic geometers, but I'd like to know about approaches different from those.

I know it kinda appears as a very vague, imprecise question, but it would be nice to know these facts in advance, so that I'll be able to keep studying from the more 'appropriate' (for me, of course) point of view.

Thank you, bye