zbMATH — the first resource for mathematics

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)

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
Full Text: DOI
[1] Appel, K.I., On Artin groups and Coxeter groups of large type, () · Zbl 0536.20019
[2] Appel, I.I.; Schupp, P.E., Artin groups and infinite Coxeter groups, Invent. math., 72, 201-220, (1983) · Zbl 0536.20019
[3] Charney, R., Artin groups of finite type are biautomatic, Math. ann., 292, 671-683, (1992) · Zbl 0736.57001
[4] A. Chermak, \(R\)-trees, small cancellation, and convergence, Trans. Amer. Math. Soc.
[5] A. Chermak, A convergence theorem for Λ-trees, J. Algebra, in press · Zbl 0984.20014
[6] Serre, J.-P., Trees, (1980), Springer-Verlag New York
[7] Tits, J., Sur le groupe des automorphisms d’un arbre, () · Zbl 0214.51301
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.