Artin groups and infinite Coxeter groups. (English) Zbl 0536.20019

Appel, K. I.; Schupp, P. E.

Invent. Math. 72, 201-220 (1983).

A Coxeter matrix over a finite set I is a symmetric matrix with entries in \(N\cup \{\infty \},\) where \(m_{ii}=1\) for \(i\in I\) and \(m_{ij}\geq 2\) for \(i\neq j\in I.\) The Artin group for this matrix has generating set \(\{a_ i:\quad i\in I\}\) and for each pair \(i\neq j\) with \(m_{ij}<\infty\) the relation equating the alternating string of \(a_ i's\) and \(a_ j's\) of length \(m_{ij}\) beginning with \(a_ i\) to the alternating string of the same length beginning with \(a_ j\). The Coxeter group associated to this Artin group is obtained by setting the squares of the generators equal to the identity. An Artin or Coxeter group is of extralarge type if \(m_{ij}\geq 4\) for all \(i\neq j\). If G is an Artin or Coxeter group with matrix \({\mathcal M}\) over the set I and \(J\subseteq I\), then \(G_ J\) is the subgroup of G generated by \(\{a_ j:\quad j\in J\}\) and \({\mathcal M}_ J\) is the restriction of \({\mathcal M}\) to \(J\times J\). After a detailed discussion of the history of these groups and their relationships with other groups, the authors use small cancellation theory to prove the following four theorems: Theorem 1. Let G be an Artin or Coxeter group of extralarge type. If \(J\subseteq I\), then \(G_ J\) has a presentation defined by the Coxeter matrix \({\mathcal M}_ J\) and the generalized word problem for \(G_ J\) in G is solvable. If \(J,K\subseteq I\), then \(G_ J\cap G_ K=G_{(J\cap K)}\). Theorem 2. An Artin group of extralarge type is torsion-free. Theorem 3. Let G be an Artin group of extralarge type. Then the set \(\{a^ 2_ i: i\in I\}\) freely generates a free subgroup of G. Theorem 4. An Artin or Coxeter group of extralarge type has solvable conjugacy problem.

Reviewer: S.C.Althoen

MSC:

20F05	Generators, relations, and presentations of groups
20F06	Cancellation theory of groups; application of van Kampen diagrams
20F10	Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
20F36	Braid groups; Artin groups
20E07	Subgroup theorems; subgroup growth

Keywords:

Artin group; small cancellation; Coxeter group of extralarge type; Coxeter matrix; generalized word problem; free subgroup; conjugacy problem

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