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Relations among the squares of the generators of the braid group. (English) Zbl 0822.20040
Let \(M=(m_{ij})\) denote an \(n \times n\) Coxeter matrix \((m_{ij}=m_{ji} \in \mathbb{Z}^{\geq 2} \cup \{\infty\}\) for \(i \neq j\) and \(m_{ii}=1\)). Let \(A(M)\) be an Artin group with generating set \(\{a_ 1, \dots, a_ n\}\) and for each \(i \neq j\) a relation \(a_ i a_ j a_ i \dots=a_ j a_ i a_ j \dots\), where both sides are words of length \(m_{ij}\). If \(m_{ij}=2\) then \(a^ 2_ i\) and \(a^ 2_ j\) commute. The Tits conjecture [stated in: K. I. Appel and P. E. Schupp, Invent. Math. 72, 201-220 (1983; Zbl 0536.20019)] says that for an arbitrary Artin group the only relations between the \(a^ 2_ i\) are these obvious commutator relations. The author proves the Tits conjecture for the braid groups.

MSC:
20F36 Braid groups; Artin groups
20F05 Generators, relations, and presentations of groups
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References:
[1] [B] Birman, J.S.: Braids, Links and Mapping Class Groups. (Ann. Math. Stud., vol. 72) Princeton, NJ: Princeton University Press 1974
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