I know there are text books of Algebraic topology. There are books of Differential geometry. But when I read papers, for example lots of papers talking about fundamental groups or higher homotopy groups of certain manifolds, sometimes lots of terminologies from abstract algebra pop out - nilpotent, solvable or amenable etc. I can understand those definitions, but I feel very uncomfortable that I don't have a geometric feeling of those languages. 
So I ask for a good reference, ideally written by geometer, that covers the material of this part?


Thanks in advance for any suggestions. 



Any text book or lecture notes regarding the algebraic part of geometry?
What about de la Harpe's topics in geometric group theory?


