Limit groups over coherent right-angled Artin groups

Abstract

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.