# Generalized pure braid groups

Generalized braid groups are by definition obtained from finite Coxeter groups after removing some of its relations. For example the Coxeter group $H_3=< a_1, a_2, a_3 | a_i^2=1, a_1a_3=a_3a_1, a_1a_2a_1=a_2a_1a_2,$ $a_2a_3a_2a_3a_2=a_3a_2a_3a_2a_3>$ and the generalized braid group $BH_3$ corresponding to $H_3$ has the above presentation after removing the relation $a_i^2=1$ for $i=1,2,3$. The kernel of the natural homomorphism $BH_3\to H_3$ is called generalized pure braid group $PBH_3$. I know the pure braid groups corresponding to some of the finite Coxeter groups can be obtained as iterated semidirect product of free groups (as they are obtained as the fundamental group of fiber-type hyperplane arrangement in the complex space). For example the classical Artin pure braid group (where the associated Coxeter group is the finite symmetric group) is of this type. My question is about the other generalized pure braid groups like, $PBH_3$, $PBH_4$, $PBD_n$ etc. Do these groups have some kind of generalized free product structure of nontrivial nature? I mean, do they act on trees with easier stabilizers (like free groups, or Abelian groups etc)?

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Probably I did not pose the question well. – Raja Mar 8 '13 at 5:09
Raja, I think your 'generalized braid groups' are more usually called 'Artin groups of finite type'. (Your question is not very clear about their definition.) Regarding your question, I'm no expert, but as far as I know the following famous paper of Deligne is the state of the art for the structure of Artin groups of finite type: Deligne, Pierre Les immeubles des groupes de tresses généralisés. (French) Invent. Math. 17 (1972), 273–302. – HJRW Mar 11 '13 at 12:32