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As we alluded to in Sect. 1.1 , associated to any Coxeter system \((W,S)\) there is a cell complex \(\Sigma \) equipped with a proper, cocompact W-action. It is the natural geometric object associated to \((W,S)\) . In this chapter we define \(\Sigma \) and describe some of its basic properties. In Sect. 8.2 of the next chapter we will prove that \(\Sigma \) is contractible and in Sect. 12.3 , that its natural piecewise Euclidean metric is \(\mathrm {CAT}(0)\) .

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The Complex \(\Sigma \)

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  • 发表日期：

    2025年

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  • Michael W. Davis

摘要

As we alluded to in Sect. 1.1 , associated to any Coxeter system \((W,S)\) there is a cell complex \(\Sigma \) equipped with a proper, cocompact W-action. It is the natural geometric object associated to \((W,S)\) . In this chapter we define \(\Sigma \) and describe some of its basic properties. In Sect. 8.2 of the next chapter we will prove that \(\Sigma \) is contractible and in Sect. 12.3 , that its natural piecewise Euclidean metric is \(\mathrm {CAT}(0)\) .