A representation of generalized braid group in classical braid group

We ask if any finite type generalized braid group is a subgroup of some classical Artin braid group. We define a natural map from a given finite type generalized braid group to a classical braid group and ask if this map is an injective homomorphism. We prove that this map is a homomorphism for the braid groups of type A_n, B_n, I_2(k). The injectivity question of this homomorphism (in these particular cases) is not yet settled. If this map is an injective homomorphism then several results will follow. For example it will follow that the Whitehead group, projective class group and the lower K-group of any subgroup of any finite type generalized braid group vanish. (For the classical braid group case this vanishing result is proved by the author and F.T. Farrell in the paper "The Whitehead groups of braid groups vanish".) Also it will follow that a finite type generalized braid group satisfies Tits alternative (recently this was asked by M. Bestvina).Comment: 22 pages, 12 figures, 1 table. it is a zipped file containing all the figure files in postscript format and the amstex source of the articl

On aspherical configuration Lie groupoids

The complement of the hyperplanes $\{x_i=x_j\}$, for all $i\neq j$ in $M^n$, for $M$ an aspherical $2$-manifold, is known to be aspherical. Here we consider the situation, when $M$ is a $2$-dimensional orbifold. We prove this complement to be aspherical for a class of aspherical $2$-dimensional orbifolds, and predict that it should be true in general also. We generalize this question in the category of Lie groupoids, as orbifolds can be identified with a certain kind of Lie groupoids.Comment: 14p. arXiv admin note: substantial text overlap with arXiv:2106.0811