The depth of a Riemann surface and of a right-angled Artin group

Abstract

We consider two families of spaces, X: the closed orientable Riemann surfaces of genus \(g>0\) and the classifying spaces of right-angled Artin groups. In both cases we compare the depth of the fundamental group with the depth of an associated Lie algebra, L, that can be determined by the minimal Sullivan algebra. For these spaces we prove that $$\begin{aligned} \text{ depth } \,{\mathbb {Q}}[\pi _1(X)] = \text{ depth }\, {L}\, \end{aligned}$$ and give precise formulas for the depth.