On Systems of Equations over Free Partially Commutative Groups

TL;DR

This paper provides an effective description and parametrization of solutions to systems of equations over right-angled Artin groups, using an analogue of Makanin-Razborov diagrams, advancing understanding of homomorphisms in these groups.

Contribution

It introduces a new method using Makanin-Razborov diagrams to describe solutions over partially commutative groups, improving previous approaches.

Findings

Effective solution set description for equations over right-angled Artin groups

Parametrization of homomorphisms from finitely generated groups to these groups

Corrections and improvements over previous normal form methods

Abstract

Version 2: Corrected Section 3.3: instead of lexicographical normal forms we now use a normal form due to V. Diekert and A. Muscholl. Consequent changes made and some misprints corrected. Using an analogue of Makanin-Razborov diagrams, we give an effective description of the solution set of systems of equations over a partially commutative group (right-angled Artin group) $G$. Equivalently, we give a parametrisation of $Hom(H, G)$, where $H$ is a finitely generated group.