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After being an open question for sixty years the Tarski conjecture
was answered in the affirmative by Olga Kharlampovich and Alexei
Myasnikov and independently by Zlil Sela. Both proofs involve long
and complicated applications of algebraic geometry over free groups
as well as an extension of methods to solve equations in free
groups originally developed by Razborov. This book is an
examination of the material on the general elementary theory of
groups that is necessary to begin to understand the proofs. This
material includes a complete exposition of the theory of fully
residually free groups or limit groups as well a complete
description of the algebraic geometry of free groups. Also included
are introductory material on combinatorial and geometric group
theory and first-order logic. There is then a short outline of the
proof of the Tarski conjectures in the manner of Kharlampovich and
Myasnikov.
The development of algebraic geometry over groups, geometric group
theory and group-based cryptography, has led to there being a
tremendous recent interest in infinite group theory. This volume
presents a good collection of papers detailing areas of current
interest.
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