Teories de primer ordre i els problemes de Tarski

A principis del segle xx, les matemàtiques varen viure una crisi de fonaments coneguda com a Grundlagenkrise der Mathematik. Com a resposta a la necessitat de formalització de les matemàtiques, la lògica matemàtica va experimentar un desenvolupament profund. Aquest desenvolupament va derivar en el naixement de diverses branques de les matemàtiques, entre les quals la teoria de models, que estudia les estructures algebraiques des de la perspectiva de la lògica matemàtica. En aquest article presentem aquest punt de vista, mostrant tant la seva potència com les seves limitacions. Comencem amb l?estudi del cos dels nombres complexos i dels nombres reals revisant els teoremes clàssics de Tarski. Continuem presentant alguns resultats de teoria de grups, com els teoremes de Szmielew sobre la teoria de primer ordre dels grups abelians. Acabem amb un resum de la solució recent dels problemes de Tarski sobre la teoria elemental dels grups lliures.At the beginning of the 20th century, mathematics suffered a foundational crisis, known as the Grundlagenkrise der Mathematik. To answer the need of formalization of mathematics, mathematical logic underwent a profound development. An important outcome of this development was the birth of a new branch of mathematics - Model Theory, which studies algebraic structures from the viewpoint of mathematical logic. In this article we present this approach, its power and its limitations. We begin by reviewing some classical results, such as theorems of A. Tarski on the fields of complex and real numbers. We then present several grouptheoretic results, including results of W. Szmielew on the first-order theory of abelian groups. We finish with a survey of the recent solution of the Tarski problems on the elementary theory of the free group. This article is based on the talk presented by the author to the XIII Meeting of the Catalan Mathematical Society

Embeddings between partially commutative groups: two counterexamples

In this note we give two examples of partially commutative subgroups of partially commutative groups. Our examples are counterexamples to the Extension Graph Conjecture and to the Weakly Chordal Conjecture of Kim and Koberda, \cite{KK}. On the other hand we extend the class of partially commutative groups for which it is known that the Extension Graph Conjecture holds, to include those with commutation graph containing no induced $C_4$ or $P_3$. In the process, some new embeddings of surface groups into partially commutative groups emerge.Comment: 15 pages, 5 figures; to appear in Journal of Algebr

On Systems of Equations over Free Partially Commutative Groups

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.Comment: 117 pages, 22 figure

Limit groups over coherent right-angled Artin groups

A new class of groups $\mathcal{C}$, containing all coherent RAAGs and all toral relatively hyperbolic groups, is defined. It is shown that, for a group $G$ in the class $\mathcal{C}$, the $\mathbb{Z}[t]$-exponential group $G^{\mathbb{Z}[t]}$ may be constructed as an iterated centraliser extension. Using this fact, it is proved that $G^{\mathbb{Z}[t]}$ is fully residually $G$ (i.e. it has the same universal theory as $G$) and so its finitely generated subgroups are limit groups over $G$. If $\mathbb{G}$ is a coherent RAAG, then the converse also holds - any limit group over $\mathbb{G}$ embeds into $\mathbb{G}^{\mathbb{Z}[t]}$. Moreover, it is proved that limit groups over $\mathbb{G}$ are finitely presented, coherent and CAT$(0)$, so in particular have solvable word and conjugacy problems.Comment: 40 pages, 1 figur

Pro-C\mathcal{C} RAAGs

Let $\mathcal{C}$ be a class of finite groups closed under taking subgroups, quotients, and extensions with abelian kernel. The right-angled Artin pro-$\mathcal{C}$ group $G_\Gamma$ (pro-$\mathcal{C}$ RAAG for short) is the pro-$\mathcal{C}$ completion of the right-angled Artin group $G(\Gamma)$ associated with the finite simplicial graph $\Gamma$. In the first part, we describe structural properties of pro-$\mathcal{C}$ RAAGs. Among others, we describe the centraliser of an element and show that pro-$\mathcal{C}$ RAAGs satisfy the Tits' alternative, that standard subgroups are isolated, and that 2-generated pro-$p$ subgroups of pro-$\mathcal{C}$ RAAGs are either free pro-$p$ or free abelian pro-$p$. In the second part, we characterise splittings of pro-$\mathcal{C}$ RAAGs in terms of the defining graph. More precisely, we prove that a pro-$\mathcal{C}$ RAAG $G_\Gamma$ splits as a non-trivial direct product if and only if $\Gamma$ is a join and it splits over an abelian pro-$\mathcal{C}$ group if and only if a connected component of $\Gamma$ is a complete graph or it has a complete disconnecting subgraph. We then use this characterisation to describe an abelian JSJ decomposition of a pro-$\mathcal{C}$ RAAG, in the sense of Guirardel and Levitt