Model theory and algebraic geometry in groups, non-standard actions and algorithmic problems

Olga Kharlampovich, Alexei Myasnikov

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

7 Scopus citations

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.

Original languageEnglish
Title of host publicationInvited Lectures
EditorsSun Young Jang, Young Rock Kim, Dae-Woong Lee, Ikkwon Yie
Pages223-245
Number of pages23
ISBN (Electronic)9788961058056
StatePublished - 2014
Event2014 International Congress of Mathematicans, ICM 2014 - Seoul, Korea, Republic of
Duration: 13 Aug 201421 Aug 2014

Publication series

NameProceeding of the International Congress of Mathematicans, ICM 2014
Volume2

Conference

Conference2014 International Congress of Mathematicans, ICM 2014
Country/TerritoryKorea, Republic of
CitySeoul
Period13/08/1421/08/14

Keywords

  • Free group
  • Group actions
  • Model theory

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