The genus invariant for Artin groups

Moroni D., Salvetti M., Villa A.

Artin groups  Coxeter Groups  Schwartz genus  Computational topology  Spectral sequences 

Let (W; S) be a Coxeter system, S finite, and let G_W be the associated Artin group. One has conguration spaces Y; Y_W; where G_W = PI_1(Y_W); and a natural W-covering f_W : Y --> Y_W. We consider the Schwarz genus g(f_W) of this covering, which is a natural topological in- variant of the Artin group. Let K = K(W; S) be the simplicial scheme of all subsets J subset of S such that the parabolic group W_J is finite. We introduce the class of Artin groups, which includes affine-type Artin groups, for which dim(K) equals the homological dimension of K; and we show that g(f_W) is always the maximum possible for this class of groups. Such maximum is given by dim(X_W) + 1; where X_W (subset of Y_W) is a CW-complex which has the same homotopy type. This result extends a previous result in [Deconcini Salvetti 2000] obtained for all finite-type Artin groups, with the exception of case A_n (for which see [Deconcini Procesi Salvetti 2004]).

Source: ISTI Technical reports, 2012