Equations and fully residually free groups

TL;DR

This paper discusses the elimination process for solving equations in free groups, leading to a structure theorem for finitely generated fully residually free groups, based on the Makanin-Razborov process.

Contribution

It introduces a detailed explanation of the elimination process and applies it to derive structural results for fully residually free groups.

Findings

Elimination process for equations in free groups explained.

Structure theorem for finitely generated fully residually free groups established.

Application of Makanin-Razborov process to group equations.

Abstract

This paper represents notes of the mini-courses given by the authors at the GCGTA conference in Dortmund (2007), Ottawa-Saint Sauveur conference (2007), Escola d'Algebra in Rio de Janeiro (2008) and Alagna (Italy, 2008) conference on equations in groups. We explain here the Elimination process for solving equations in a free group which has Makanin-Razborov process as a prototype. We also explain how we use this process to obtain the structure theorem for finitely generated fully residually free groups and many other results.