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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.

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You can go very very far with not much more than a sensible textbook on "abstrac algebra" like Herstein's, Lang's or —in fact—many others. – Mariano Suárez-Alvarez Aug 21 '12 at 21:31
In any case it would probably be good if you were more explicit about what exactly you are interested in subjectwise: for otherwise not a lot can be said apart from «general algebra textbook»! – Mariano Suárez-Alvarez Aug 21 '12 at 21:33
Perhaps "Commutative Algebra: with a View Toward Algebraic Geometry" by Eisenbud is what you are after? – Drew Heard Aug 21 '12 at 22:57
Note that "amenable" is really a geometric condition, not an algebraic one. – Paul Siegel Aug 21 '12 at 23:57
Drew: The OP seems to be interested in group theoretic terms. Commutative algebra is mostly disjoint from group theory, at least at this level. – Charles Staats Aug 22 '12 at 2:13
up vote 5 down vote accepted

What about de la Harpe's topics in geometric group theory?

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Are you talking of the book edited by Ghys and de la Harpe? – Benoît Kloeckner Aug 22 '12 at 11:15
No, I am talking at PdlH's topics in geometric group theory. – Igor Rivin Aug 22 '12 at 13:18

As you mentioned higher homotopy groups, it's worth noting that these are always abelian, which is a stronger condition than each of nilpotent, solvable and amenable. So you probably won't find these adjectives used when describing higher homotopy groups.

Fundamental groups need not be abelian, but they may be almost abelian, and each of the three properties you mention give different ways of formalizing what this should mean. As Igor mentions in his answer, the study of group properties via their realization as fundamental groups of complexes is the topic of geometric group theory.

As for references, I do not know the geometry literature all that well, but Chapter 10 of G. W. Whitehead's book "Elements of Homotopy Theory" contains a nice introductions to nilpotency for topologists.

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