@inproceedings{2af826bd7e16417199b5f861a926db51,
title = "Model theory and algebraic geometry in groups, non-standard actions and algorithmic problems",
abstract = "We discuss the modern theory of equations in groups, algebraic geometry and model theory in free and hyperbolic groups, as well as group actions on-trees. One of our main tools is a combinatorial process that combines and generalizes a number of known results and algorithms, such as the Makanin-Razborov process for solving equations in groups, Rauzy-Veech induction in dynamical systems, classification of basic group actions in group theory and topology, and elimination and parametrization theorems in classical algebraic geometry. The development of algebraic geometry comes together with advances in the theory of fully residually free and fully residually hyperbolic groups, which are coordinate groups of irreducible algebraic varieties. We describe finitely generated groups elementarily equivalent to a free non-abelian group (another classification is given by Sela) and show that the first-order theory of a free or a torsion-free hyperbolic group is decidable (solution to Tarski's problems from 1940's). Furthermore, for such groups we give an algorithm for elimination of quantifiers to boolean combinations of-formulas. We also provide a description of definable sets in a torsion-free hyperbolic group (in particular, in a free group) and demonstrate that only cyclic subgroups and the whole group are definable in these groups (this solves Malcev's problem of 1965). In the group actions section we describe all finitely presented groups acting freely on-trees (solution to Alperin's and Bass problem of 1990). At the end we outline some related open problems.",
keywords = "Free group, Group actions, Model theory",
author = "Olga Kharlampovich and Alexei Myasnikov",
note = "Publisher Copyright: {\textcopyright} 2014 Proceeding of the International Congress of Mathematicans, ICM 2014. All rights reserved.; 2014 International Congress of Mathematicans, ICM 2014 ; Conference date: 13-08-2014 Through 21-08-2014",
year = "2014",
language = "English",
series = "Proceeding of the International Congress of Mathematicans, ICM 2014",
pages = "223--245",
editor = "Jang, {Sun Young} and Kim, {Young Rock} and Dae-Woong Lee and Ikkwon Yie",
booktitle = "Invited Lectures",
}