CAT(0) metric for exceptional almost spherical types
Show that for any Artin group A_S and any almost spherical subset S'⊂S of type in {\widetilde F_4, \widetilde E_6, \widetilde E_7, \widetilde E_8, [3,5,3], [5,3,3,3]}, the relative Artin complex Δ_{S,S'} equipped with the piecewise Euclidean or piecewise hyperbolic metric induced from a fundamental domain E_{S'} of the Coxeter group W_S acting on X∈{R^n, H^n} is CAT(0) and therefore contractible.
References
Conjecture. Let $A_S$ be an Artin group and suppose $S$ contains a subset $S'$ which is almost spherical, and its type is contained in ${\widetilde F_4,\widetilde E_6,\widetilde E_7,\widetilde E_8,[3,5,3],[5,3,3,3]}$. Then the associated Coxeter group $W_S$ acts properly and cocompactly by isometries on $X$, with $X=\mathbb En$ or $\mathbb Hn$. Let $E_{S'}$ be a fundamental domain of $W_S\curvearrowright X$ equipped with the metric inherited from the standard metric on $X$. We metrize the fundamental domain $K_{S'}$ of $\Delta_{S,S'}$ using the metric on $E_{S'}$ as in Definition~\ref{def:metric2}, which gives a piecewise Euclidean or piecewise hyperbolic metric on $\Delta_{S,S'}$. Then $\Delta_{S,S'}$ equipped with such metric is CAT$(0)$, hence contractible.