Introductory text on geometric group theory? Can someone indicate me a good introductory text on geometric group theory? 
 A: 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.
A: 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/coll-63/
A: 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 Bass-Serre 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: For beginner, I think these notes are very good.. http://www.mathematik.uni-regensburg.de/loeh/teaching/ggt_ws1011/lecture_notes_old.pdf
A: 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 quasi-geometry, such as growth, hyperbolicity, boundary constructions and amenability.
A: 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 Non-Positive 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 "Non-Positive 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 well-written and pretty gentle.
A: 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.
A: 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.
A: discrete groups by Kenichi Oshika is one of my favorites
A: For beginners 'A Course on Geometry Group Theory' by Brian Bowditch is nice although quite terse in places, but good for 'setting the scene'.
A: 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).
A: 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 non-trivial results, including a fairly complete review of the Grigorchuk group.
A: What about this wonder:
Peter Scott, Terry Wall, Topological methods in group theory, London Math. Soc. Lecture Note Ser., 36, Cambridge Univ. Press (1979) 137-203.
simply beauty and useful
A: 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.)
