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.