The Tarski Problems and Their Solutions

Benjamin Fine^1*, Anthony Gaglione², Gerhard Rosenberger³, Dennis Spellman⁴

¹Department of Mathematics Fairfield University, Fairfield, USA.
²Department of Mathematics, United States Naval Academy, Annapolis, USA.
³Department of Mathematics, University of Hamburg, Hamburg, Germany.
⁴Department of Mathematics, Temple University, Philadelphis, USA.

DOI: 10.4236/apm.2015.54023                

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 has been actually proved. We then discuss the history of the solution as well as the components of the proof. We then provide the basic strategy for the proof. We finish this paper with a brief discussion of elementary free groups.

Keywords

Non-Abelian Free Group, Elementary Theory, Tarski Problems, Elementary Free Groups, Algebraic Geometry over Groups

Conflicts of Interest

The authors declare no conflicts of interest.

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