On Tarski's Decidability Problem

TL;DR

This paper reviews the current state of Tarski's decidability problems, contrasting combinatorial and topological approaches, and corrects misconceptions and errors in recent literature by Z. Sela.

Contribution

It clarifies distinctions between different approaches to Tarski's problems and provides corrections to recent misstatements and errors in Sela's work.

Findings

Identifies errors in Sela's papers on Diophantine Geometry.

Highlights differences between combinatorial and topological approaches.

Corrects misconceptions about the relations between various concepts in the field.

Abstract

This note provides a brief guide to the current state of the literature on Tarski's problems with emphasis on features that distinguish the approach based on combinatorial and algorithmic group theory from the topological approach to Tarski's problem. We use this note to provide corrections to some typos and to address some misconceptions from the recent report by Z. Sela about the relations between the concepts and results in the approaches to the Tarski problems. We were forced to read Sela's papers to be able to address some of his comments, and found errors in his papers 6, 3 and 4 on Diophantine Geometry published in GAFA and Israel J. Math. which we mention in Section 4. His proceedings of the ICM 2002 paper also contains wrong Theorem 6 (to make it correct one has to change the definition of non-elementary hyperbolic $\omega$-residually free towers to make them equivalent to our coordinate groups of regular NTQ systems.)