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Locally non-spherical Artin groups. (English) Zbl 0901.20025
Let $$S$$ be a finite nonempty set, and let $$M=(m_{a,b})_{a,b\in S}$$ be a Coxeter matrix over $$S$$. For $$m$$ in $$\mathbb{N}\cup\{\infty\}$$ and $$a,b$$ in $$S$$, let $$w(a,b;m)$$ denote $$(ab)^{{m\over 2}}$$ if $$m$$ is even, $$(ab)^{{m-1\over 2}}a$$ if $$m$$ is odd, and $$1$$ if $$m$$ is $$\infty$$. The Artin group associated with $$M$$ is the group $$G(M)$$ presented by $$S$$ subject to the relations $$w(a,b;m_{a,b})=w(b,a;m_{a,b})$$ for all $$a,b$$ in $$S$$. We say further that $$M$$ and $$G(M)$$ are locally non-spherical if ${1\over m_{a,b}}+{1\over m_{a,c}}+{1\over m_{b,c}}\leq 1\quad\text{for all } a,b,c\text{ in }S,\;a\neq b\neq c\neq a.$ Consider a locally non-spherical Artin group $$G$$. The author exhibits an explicit solution of the word problem for $$G$$ based on the study of a certain set of hyperbolic isometries of a tree. Consider a nonempty subset $$T$$ of $$S$$, let $$M_T$$ be $$(m_{a,b})_{a,b\in T}$$, and let $$G_T$$ be the subgroup of $$G$$ generated by $$T$$. The author also proves that $$G_T$$ is naturally isomorphic to the Artin group associated with $$M_T$$.
Reviewer: L.Paris (Dijon)

##### MSC:
 20F36 Braid groups; Artin groups 20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) 20F05 Generators, relations, and presentations of groups 20F55 Reflection and Coxeter groups (group-theoretic aspects) 20E08 Groups acting on trees
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##### References:
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