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 $\mathrm{\mathbb{Z}}\left[t\right]$-exponential group $G^{\mathrm{\mathbb{Z}}\left[t\right]}$ may be constructed as an iterated centraliser extension. Using this fact, it is proved that $G^{\mathrm{\mathbb{Z}}\left[t\right]}$ 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 ${\mathbb{G}}^{\mathrm{\mathbb{Z}}\left[t\right]}$. Moreover, it is proved that limit groups over $\mathbb{G}$ are finitely presented, coherent and $CAT\left(0\right)$, so in particular have solvable word and conjugacy problems.