The Tarski Theorems and Elementary Free Groups (Algebras, logics, languages and related areas)

Abstract

Around 1945, Alfred Tarski proposed several questions concerning the elementary theory of non-abelian free groups. These remained open for 60 years until they were proved by O. Kharlampovich and A. Myasnikov and independently by Z. Sela. The proofs, by both sets of authors, were monumental and involved the development of several new areas of infinite group theory. In this paper we explain precisely the Tarski problems and what was actually proved. We then discuss the history of the solution as well the components of the proof and provide the basic startegy for the proof. We finish with a brief discussion of elementary free groups, that is groups that have exactly the same elementary theory as the class of nonabelian free groups