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.

中文翻译：

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黎曼曲面和直角Artin组的深度

我们考虑两个空间家族X：属\（g> 0 \）的闭合可定向Riemann曲面和直角Artin组的分类空间。在这两种情况下，我们都将基本群的深度与关联的李代数L的深度进行比较，后者可以由最小的沙利文代数确定。对于这些空间，我们证明$$ \ begin {aligned} \ text {depth} \，{\ mathbb {Q}} [\ pi _1（X）] = \ text {depth} \，{L} \，\ end {aligned} $$并给出深度的精确公式。