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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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