Representation varieties of RAAGs

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.

💡 Research Summary

The paper investigates the topology of representation varieties associated to right‑angled Artin groups (RAAGs) when the target is a Lie group G. For a finite simplicial graph K, the associated RAAG Γ_K is generated by a vertex for each node of K with commutation relations imposed on pairs of generators corresponding to edges. The G‑representation variety R(K,G)=Hom(Γ_K,G) can be identified with the subset of G^V consisting of tuples (g_v)_{v∈V} such that