The Tits conjecture for locally reducible Artin groups (Q2709982)

Given a finite set \(S\) and, for each pair \((s,t)\in S\times S\) a number \(m_{s,t}=m_{t,s}\in\{1,2,3,\dots,\infty\}\) (with \(m_{s,t}=1\) iff \(s=t\)), the group \(A\) with presentation NEWLINE\[NEWLINE\langle S\mid stst\cdots=tsts\cdots\text{ for each }(s,t)\in S\times S\rangleNEWLINE\]NEWLINE (where the expressions \(stst\cdots\) and \(tsts\cdots\) each have length \(m_{s,t}\)) is called an `Artin group'. The set \(S\) is called the `standard generating set' of \(A\). The most familiar examples of Artin groups are the classical braid groups.NEWLINENEWLINENEWLINELet \(A\) be an Artin group with standard generating set \(S\). For each \(s\in S\), let \(T_s=s^2\), and let \(H\) be the subgroup of \(A\) generated by \(\{T_s\mid s\in S\}\). Clearly \(T_sT_t=T_tT_s\) if \(m(s,t)=2\); the conjecture of Tits mentioned in the title says that the set \(\{T_sT_t=T_tT_s\mid(s,t)\in S\times S,\;m_{s,t}=2\}\) forms a complete set of defining relations for \(H\). Before this paper, the conjecture had been proved in a variety of special cases, notably the braid groups.NEWLINENEWLINENEWLINEIn this paper, the Tits conjecture is proved for a larger class of Artin groups -- the so-called `locally reducible' ones -- which includes many of the special cases mentioned above. The proof relies on the fact that a certain complex, called the `Deligne complex', associated to a locally reducible Artin group has a CAT(0) geometry.NEWLINENEWLINENEWLINERemark: After this paper was published, the Tits conjecture was proved for all Artin groups by \textit{J. Crisp} and \textit{L. Paris} [Invent. Math. 145, No. 1, 19-36 (2001; Zbl 1002.20021)].