Can someone indicate me a good introductory text on geometric group theory?

1$\begingroup$ There are several directions in geometric group theory: orientation to lowdimensional topology, study of very general presentations, small cancelation, focus on some class (Coxeter, hyperbolic...) or topic (Dehn function). It would be a benefit if the answers were including more information about this. Many pairs of books listed below have essentially disjoint contents. $\endgroup$– YCorJan 9, 2018 at 13:28
14 Answers
de la Harpe's book is quite nice and has an amazing bibliography, but it doesn't really prove any deep theorems (though it certainly discusses them!). Some other sources.
1) Bridson and Haefliger's book "Metric Spaces of NonPositive Curvature". Very easy to read and covers a lot of ground.
2) Ghys and de la Harpe's book on hyperbolic groups. Another classic, but in French. If you look around the web, you can find English translations.
3) Cannon's survey "Geometric Group Theory" in the Handbook of Geometric Topology is very nice.
4) Bowditch's survey "A course on geometric group theory" is also very nice.
5) Bridson has written two beautiful surveys entitled "NonPositive Curvature in Group Theory" and "The Geometry of the Word Problem". The latter was one of the first things I read in any depth.
6) Geoghegan's "Topological Methods in Group Theory" is very nice, with a more topological approach.
7) Mike Davis's "The Geometry and Topology of Coxeter Groups" is a bit specific, but covers a lot of important material in a nice way.
8) John Meier's book "Groups, Graphs and Trees: An Introduction to the Geometry of Infinite Groups" is wellwritten and pretty gentle.
A classic, but perhaps not as "geometric" as contemporary sources, is Lyndon and Schupp's Combinatorial Group Theory (named after the classic Combinatorial Group Theory, by Magnus, Karrass, and Solitar).

7$\begingroup$ Lyndon and Schupp is really the only textbook source for a lot of classical material on free groups. And MagnusKarrassSolitar is the only textbook source for the "Magnus representation" of a free group, and thus for the residual nilpotence of free groups. However, I think both sources are hard to read for someone whose mind works in geometric ways (that might be why I secretly like them a lot!) $\endgroup$ Nov 2, 2009 at 23:49

3$\begingroup$ Also, MagnusKarrassSolitar might be the only book ever written which is hard to read because the authors included too many details in their proofs! $\endgroup$ Nov 2, 2009 at 23:50

3$\begingroup$ MagnusKarrassSolitar has lots of gems. That's where I learned that ascending unions of free groups of bounded rank aren't free. $\endgroup$ Nov 3, 2009 at 3:09

4$\begingroup$ I feel obligated to mention that if anyone ever needs to learn about Whitehead's algorithm, it's better to read Whitehead's original paper (or Stalling's "Whitehead graphs on handlebodies") than to read Lyndon and Schupp's account. Sometimes they took really topological things and made them too algebraic. $\endgroup$ Nov 3, 2009 at 3:11

3$\begingroup$ You can even give an easy example of ascending unions of free groups not being free  Q is an ascending union of groups each of which is isomorphic to Z! Another warning one should make about Lyndon and Schupp is that there are an astounding number of errors/misprints. $\endgroup$ Nov 3, 2009 at 4:41
Pierre de la Harpe's "Topics in Geometric Group Theory" is, to be fair, the only book I know relatively well so I can't compare it to others. Anyway, I do like it  the writing style is pleasant and it gets to some nontrivial results, including a fairly complete review of the Grigorchuk group.
For a very introductory text to a wide range of subfields, there is Office Hours with a Geometric Group Theorist, edited by Matt Clay and Dan Margalit. (Here is the table of contents for a quick peek.)
What about this wonder:
Peter Scott, Terry Wall, Topological methods in group theory, London Math. Soc. Lecture Note Ser., 36, Cambridge Univ. Press (1979) 137203.
simply beauty and useful
There is a very nice book related to the topic  "Word processing in groups" by David Epstein. It covers some stuff about the combinatorial aspects of geometric group theory, e.g. automatic groups, combable groups etc.

1$\begingroup$ That book (according to google) is actually by Epstein, Paterson, Camon, Holt, Levy, and Thurston. It's usually referred to as Epstein, et. al. $\endgroup$ Jun 29, 2010 at 5:03

1$\begingroup$ The title page says "Epstein, with Cannon, Holt, Levy, Paterson Thurston". The cover juxtaposes the names, but with a bigger "Epstein" and then smaller "Cannon, Holt", etc. So "Epstein et alii" doesn't seem a bad way to reference this book, but I guess one could write "Epstein cum aliis" instead :) $\endgroup$ Apr 14, 2011 at 9:08
Still missing in this list, to appear in March 2018, but with some earlier versions online, and very comprehensive:
Drutu & Kapovich: „Geometric Group Theory“, http://bookstore.ams.org/coll63/
A book I quite like is Bogopolski's Introduction to Group Theory. It's not really an introduction (at least at undergraduate level), but it covers some things that aren't covered in the books above, particularly automorphisms of free groups and it has more BassSerre theory than anything I've read that's mentioned in the other answers.
I also want to add a dissenting opinion on de la Harpe's book. I think it's quite disappointing, given that it's the first real textbook since geometric group theory went beyond combinatorial group theory.
A book I completely appreciate is "Geometric Group Theory", by Clara Loh.
It contains 250 exercises of varying difficulty including programming tasks. She introduces the key notions from quasigeometry, such as growth, hyperbolicity, boundary constructions and amenability.
For beginner, I think these notes are very good.. http://www.mathematik.uniregensburg.de/loeh/teaching/ggt_ws1011/lecture_notes_old.pdf
My personal favorite learner (not a reference) for geometric group theory is John Stalling's notes from a course that he gave at Berkeley about a decade ago. It's terse, since they are just lecture notes, but I like his style of exposition and there are many examples to work through in the exercises, which I found helpful.
It's not an introductory text, but if you're trying to get a feel for the area you could look at the GGT Open Problems Wiki. It's still rather incomplete and patchy; a more coherent and shorter alternative is Bestvina's Problem List.
discrete groups by Kenichi Oshika is one of my favorites
For beginners 'A Course on Geometry Group Theory' by Brian Bowditch is nice although quite terse in places, but good for 'setting the scene'.