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# memo_has_moved_text();On systems of equations over free partially commutative groups

Montserrat Casals-Ruiz, Montserrat Casals-Ruiz, Department of Mathematics and Statistics, McGill University, 805 Sherbrooke St. West, Montreal, Quebec H3A 2K6, Canada and Ilya Kazachkov, Ilya Kazachkov, Department of Mathematics and Statistics, McGill University, 805 Sherbrooke St. West, Montreal, Quebec H3A 2K6, Canada

Publication: Memoirs of the American Mathematical Society
Publication Year: 2011; Volume 212, Number 999
ISBNs: 978-0-8218-5258-3 (print); 978-1-4704-0616-5 (online)
DOI: http://dx.doi.org/10.1090/S0065-9266-2010-00628-8
Published electronically: December 22, 2010
MathSciNet review: 2817144
Keywords:Equations in groups, partially commutative group, right-angled Artin group, Makanin-Razborov diagrams, algebraic geometry over groups
MSC: Primary 20F70; Secondary 20F10, 20F36

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Chapters

• Chapter 1. Introduction
• Chapter 2. Preliminaries
• Chapter 3. Reducing systems of equations over $\mathbb {G}$ to constrained generalised equations over $\mathbb {F}$
• Chapter 4. The process: Construction of the tree T
• Chapter 5. Minimal solutions
• Chapter 6. Periodic structures
• Chapter 7. The finite tree $T_0(\Omega )$ and minimal solutions
• Chapter 8. From the coordinate group $\mathbb {G}_{R(\Omega ^*)}$ to proper quotients: The decomposition tree $T_{\mathrm {dec}}$ and the extension tree $T_{\mathrm {ext}}$
• Chapter 9. The solution tree $T_{\mathrm {sol}}(\Omega )$ and the main theorem

### Abstract

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) $\mathbb{G}$. Equivalently, we give a parametrisation of $\mathrm{Hom}(G, \mathbb{G})$, where $G$ is a finitely generated group.