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I would please like some recommendations for an introductory level book on combinatorial group theory, by which I mean a group theory book which places emphasis on generators and relations and free groups, and then discusses common concepts such as quotient groups in terms of these. Thank you.

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I would suggest the book

  • Groups, Graphs and Trees an introduction to the geometry of infinite groups by John Meier.

This is an excellent introductory text. It is well written, covers a broad range of topics in geometric and combinatorial group theory, and contains lots of examples (every second chapter is a study of an example). Also, it is modern (2008) - the other suggested texts are all 80s and earlier! (The other books are, of course, still very relevant, but Meier's book allows you to see where the subject is today, as opposed to in the pre-Gromov days.)

That said, I do not believe that you can survive in the world of combinatorial group theory without reading Magnus, Karrass and Solitar. So Meier's book is actually my secondary recommendation. Read Meier so that when you read Magnus, Karrass and Solitar you will understand it better.

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    $\begingroup$ Thanks for your recommendation and helpful comment on reading Meier before Magnus, it looks very good. $\endgroup$ – user50229 Oct 17 '14 at 10:44
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    $\begingroup$ I accepted your answer because you suggested a modern text which looks like a nice introduction, and a strategy for then reading Magnus, Karrass and Solitar, but really I would like to accept all the answers as very helpful! $\endgroup$ – user50229 Oct 24 '14 at 9:12
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As mentioned, Presentations of Groups by D.L. Johnson is nice.

Try also these books:

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    $\begingroup$ The last two, and specially the second one, are the monsters of combinatorial group theory: great, wonderful books. $\endgroup$ – Timbuc Oct 15 '14 at 4:04
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    $\begingroup$ But Lyndon and Schupp could not be described as an introductory level book! $\endgroup$ – Derek Holt Oct 15 '14 at 4:27
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    $\begingroup$ The Magnus one looks very nice, and it is Dover which is another plus. $\endgroup$ – user50229 Oct 15 '14 at 12:06
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I would recommend "Presentations of Groups" by D.L. Johnson. It is good as an introduction to free groups and group presentations, but does assume a basic knowledge of group theory. Since a group defined by a presentation is defined as a quotient group of a free group, you need to have some basic familiarity with quotient groups and the isomorphism theorems before starting to learn about group presentations.

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  • $\begingroup$ Thanks, I did not see this book when I searched. $\endgroup$ – user50229 Oct 15 '14 at 12:08
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    $\begingroup$ @user50229 That is understandable - I once spent an evening trying to persuade two other (much more experienced, and well-known) geometric group theorists that "someone called D. Johnson wrote a book called Presentations of Groups". They were trying to persuade me that "the geometric group theorist called D. Johnson, the one with the homomorphism named after him, is too much of a hippy to sit down and write a book." I never found out if the two Johnson's were the same...Also, I would argue that Magnus, Karrass and Solitar is "standard". I mean - my supervisor read it when he was starting! $\endgroup$ – user1729 Oct 15 '14 at 12:51
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I would also recommend the book Trees of Jean-Pierre Serre, which has a highly original and elegant approach to combinatorial and geometrical group theory.

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