# The genus of the configuration spaces for Artin groups of affine type

### Davide Moroni

National Research Council of Italy (CNR), Pisa, Italy### Mario Salvetti

Università di Pisa, Italy### Andrea Villa

ISI Foundation, Torino, Italy

## Abstract

Let $(\mathbf W,S)$ be a Coxeter system, $S$ finite, and let $\mathbf G_{\mathbf W}$ be the associated Artin group. One has {\it configuration spaces} $\mathbf Y,\ \mathbf Y_{\mathbf W},$ where $\mathbf G_{\mathbf W}=\pi_1(\mathbf Y_{\mathbf W}),$ and a natural $\mathbf W$-covering $f_{\mathbf W}:\ \mathbf Y\to\mathbf Y_{\mathbf W}.$ The {\it Schwarz genus} $g(f_{\mathbf W})$ is a natural topological invariant to consider. In \cite{salvdec2} it was computed for all finite-type Artin groups, with the exception of case $A_n$ (for which see \cite{vassiliev},\cite{salvdecproc3}). In this paper we generalize this result by computing the Schwarz genus for a class of Artin groups, which includes the affine-type Artin groups. Let $K=K(\mathbf W,S)$ be the simplicial scheme of all subsets $J\subset S$ such that the parabolic group $\mathbf W_J$ is finite. We introduce the class of groups for which $dim(K)$ equals the homological dimension of $K,$ and we show that $g(f_{\mathbf W})$ is always the maximum possible for such class of groups. For affine Artin groups, such maximum reduces to the rank of the group. In general, it is given by $dim(\mathbf X_{\mathbf W})+1,$ where $\mathbf X_{\mathbf W}\subset \mathbf Y_{\mathbf W}$ is a well-known $CW$-complex which has the same homotopy type as $\mathbf Y_{\mathbf W}.

## Cite this article

Davide Moroni, Mario Salvetti, Andrea Villa, The genus of the configuration spaces for Artin groups of affine type. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 25 (2014), no. 3, pp. 233–248

DOI 10.4171/RLM/676