Limit groups over coherent right-angled Artin groups

TL;DR

This paper introduces a new class of groups encompassing coherent RAAGs and toral relatively hyperbolic groups, demonstrating their limit groups' properties and embedding relations, with implications for their algebraic and geometric structure.

Contribution

It defines a new class of groups and proves that limit groups over these are finitely presented, coherent, and CAT(0), establishing their algebraic and geometric properties.

Findings

Limit groups over coherent RAAGs are finitely presented.

Limit groups over these groups are coherent and CAT(0).

Embedding results relate limit groups to their parent 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.