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Artin groups of finite type with three generators. (English) Zbl 0996.20022
Introduction: Let $$W$$ be a finite Coxeter group on three generators $$A$$, $$B$$, $$C$$, and consider the set of all possible expressions of the Coxeter element $$X=BAC$$ in $$W$$ as a product of three reflections. In Section 3 we construct a 3-dimensional CW-complex, $$K(W)$$, which we can associate to this set in a natural way. (Daan Krammer has informed us that he has also considered this complex and has obtained similar results.) We show that this complex enjoys two remarkable properties. First, the fundamental group of $$K(W)$$ is isomorphic to the finite type Artin group determined by $$W$$; second, $$K(W)$$ can be given a piecewise Euclidean (PE) metric of nonpositive curvature. Thus, if $$G$$ is an Artin group of finite type with three generators, then $$G$$ acts cocompactly on a 3-dimensional PE complex of nonpositive curvature.
The paper is arranged as follows. In Section 2, we make the corresponding construction for two-generator Artin groups and prove that the associated 2-complex has nonpositive curvature. In Section 3, we define the complex $$K$$ and show that it has the correct fundamental group. In Section 4, we give $$K$$ a PE metric and show that it has nonpositive curvature.

MSC:
 20F36 Braid groups; Artin groups 20F05 Generators, relations, and presentations of groups 20F55 Reflection and Coxeter groups (group-theoretic aspects) 20F65 Geometric group theory 57M07 Topological methods in group theory
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