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 $$ \mbox{depth} \,\mathbb Q[π_1(X)] = \mbox{depth}\, {L}\,$$ and give precise formulas for the depth.