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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:
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