On the residual nilpotence of pure Artin groups.

*(English)*Zbl 1103.20035The author gives a simple proof of the residual nilpotence of pure Artin groups of spherical type, based on the generalization by F. Digne [J. Algebra 268, No. 1, 39-57 (2003; Zbl 1066.20044)] of Krammer’s faithful linear representation of the braid group, to crystallographic finite-type Artin groups. Previously, the residual nilpotence was known only for types \(A\), \(B\), and \(D\), by complicated arguments, and the rank-two groups.

Here is a sketch of the argument. Let \(W\) be a finite Coxeter group, irreducible and crystallographic, \(B\) the associated Artin group, and \(P=\ker(B\to W)\) the pure Artin group. Digne’s representation has the form \(\varphi\colon B\to\text{GL}(V\otimes\mathbb{R}[\![h]\!])\), where \(V\) is the real vector space with basis the set of reflections in \(W\). Specializing to \(h=0\) in \(\varphi\) results in the permutation representation of \(W=B/P\) on \(V\). It follows easily that, for \(x\in P\), \(\varphi(x)-1\) is divisible by \(h\), and then, by induction, that \(\varphi(x)-1\) is divisible by \(h^n\) for \(x\) in the \(n\)-th term of the lower central series of \(P\). Then the intersection of the lower central series is contained in the kernel of \(\varphi\), whence is trivial since \(\varphi\) is faithful. For irreducible, non-crystallographic groups, the author uses the “folding maps” of J. Crisp [in Geometric group theory down under, de Gruyter, 119-137 (1999; Zbl 1001.20034)], which inject the Artin group into a crystallographic Artin group.

Here is a sketch of the argument. Let \(W\) be a finite Coxeter group, irreducible and crystallographic, \(B\) the associated Artin group, and \(P=\ker(B\to W)\) the pure Artin group. Digne’s representation has the form \(\varphi\colon B\to\text{GL}(V\otimes\mathbb{R}[\![h]\!])\), where \(V\) is the real vector space with basis the set of reflections in \(W\). Specializing to \(h=0\) in \(\varphi\) results in the permutation representation of \(W=B/P\) on \(V\). It follows easily that, for \(x\in P\), \(\varphi(x)-1\) is divisible by \(h\), and then, by induction, that \(\varphi(x)-1\) is divisible by \(h^n\) for \(x\) in the \(n\)-th term of the lower central series of \(P\). Then the intersection of the lower central series is contained in the kernel of \(\varphi\), whence is trivial since \(\varphi\) is faithful. For irreducible, non-crystallographic groups, the author uses the “folding maps” of J. Crisp [in Geometric group theory down under, de Gruyter, 119-137 (1999; Zbl 1001.20034)], which inject the Artin group into a crystallographic Artin group.

Reviewer: Michael J. Falk (Flagstaff)

##### MSC:

20F36 | Braid groups; Artin groups |

20F55 | Reflection and Coxeter groups (group-theoretic aspects) |

20E26 | Residual properties and generalizations; residually finite groups |

##### Keywords:

residual nilpotence; pure Artin groups; Krammer-Digne representation; finite Coxeter groups**OpenURL**

##### References:

[1] | DOI: 10.1007/BF02785852 · Zbl 1078.20038 |

[2] | DOI: 10.1016/S0021-8693(03)00327-2 · Zbl 1066.20044 |

[3] | DOI: 10.1007/BF01394780 · Zbl 0574.55010 |

[4] | DOI: 10.1017/S000497270001995X · Zbl 0996.20023 |

[5] | DOI: 10.1007/BF00181653 · Zbl 0794.20047 |

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