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

## TL;DR

This paper compares the depth of fundamental groups of Riemann surfaces and right-angled Artin groups with associated Lie algebra depths, establishing equalities and providing explicit formulas.

## Contribution

It proves the equality of depths between fundamental groups and associated Lie algebras for these spaces, with explicit formulas for the depth.

## Key findings

- Depth of fundamental group equals depth of associated Lie algebra
- Explicit formulas for the depth of these spaces
- Applicable to Riemann surfaces and right-angled Artin groups

## 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 $$ \mbox{depth} \,\mathbb Q[\pi_1(X)] = \mbox{depth}\, {L}\,$$ and give precise formulas for the depth.