Representation varieties of RAAGs

TL;DR

This paper studies the structure of representation varieties of right-angled Artin groups into various Lie groups, revealing their connectivity properties and classifying components for specific cases like SO(3).

Contribution

It provides a comprehensive analysis of the connectivity of G-representation varieties of RAAGs for multiple Lie groups, including new classifications for SO(3) cases.

Findings

Representation varieties are connected for SU(n), Sp(n), U(n).

Representation varieties are generally not connected for SO(n), Spin(n) when n ≥ 3.

Number of components for SO(3) representations of RAAGs that are 3-manifold groups is determined.

Abstract

We investigate the $G$-representation varieties of right-angled Artin groups (RAAGs) for various Lie groups $G$. We show these varieties are connected for a large class of such $G$, including $\mathrm{SU}(n), \mathrm{Sp}(n)$ and $\mathrm{U}(n)$, while they are generally not connected for other large classes, such as $\mathrm{SO}(n)$ and $\mathrm{Spin}(n)$ for $n \geq 3$. When $G = \mathrm{SO}(3)$ we determine the number of connected components of the representation variety associated to any RAAG that is also a 3-manifold group.