Salvetti complex of dihedral Artin group

Question

The Salvetti complex of a RAAG is well-known and it is fairly simple, since each complete graph gives rise to a tori. The case of Artin groups is wilder, since we do not have tori anymore. The construction is quite abstract and there is no easy description of it, but I was hoping to be able to understand the Salvetti complex in some simpler cases. For example, if we consider the case of a dihedral Artin group, whose presentation is:

$$G_n=\left\langle u,v\,\middle\vert\,\underset{n\text{ letters}}{\underbrace{uvu\dots}}=\underset{n\text{ letters}}{\underbrace{vuv\dots}}\right\rangle$$ then the Salvetti complex of $G_n$ is equal to the presentation complex of the group. If $n=2$ we just have a RAAG and the Salvetti complex is a torus, but what about $n\geq 3$? Is there any ''simple'' description of this complex?

My objective is the following: for a general RAAG $A_\Gamma$ each vertex $v\in\Gamma$ corresponds to a circle $S^1_v$ in the Salvetti complex. Hence, for each clique $\Delta=\lbrace v_1,\dots,v_k\rbrace$ the corresponding cell in the complex is $T_\Delta=S^1_{v_1}\times\cdots\times S^1_{v_k}$. Now, consider a map $\varphi:A_\Gamma\to\mathbb{Z}$, then we have a natural map $T_\Delta\to S^1=\mathbb{R}/\mathbb{Z}$ given by: $$(x_1,\dots,x_k)\mapsto\varphi(v_1)x_1+\cdots+\varphi(v_k)x_k+\mathbb{Z}$$ and those maps extends onto a map $T_\Gamma\to S^1$, where $T_\Gamma$ denotes the Salvetti complex. What I was trying to do is to generalize this construction to find a map from the Salvetti complex of an Artin group onto $S^1$ with similar properties. However, this seems imposible in general, so that is why I was trying to consider just the case of dihedral Artin groups, which is the easiest case. More ambitiously I would like to do the construction for $2$-dimensional Artin groups, but this looks way hrader.

Answer 1

Here is a simple description of the Salvetti complex of a dihedral Artin group $G_n$: it is the presentation complex. Start with a wedge of two oriented circles, labeled u and v. In the case $n=2$, you glue in a square on this wedge, according to the boundary label uvu^{-1}v^{-1}: you obtain the $2$-torus. Similarly, for an arbitrary $n \geq 2$, you glue in a $2n$-gon on this wedge of two circles, according to the boundary label $uvu \dots \dots v^{-1}u^{-1}v^{-1}$. The Salvetti complex of an arbitrary spherical type Artin group has a similar description, using Coxeter cells.