Title: Configuration Lie groupoids and orbifold braid groups
Abstract: We propose two definitions of configuration Lie groupoids and in
both the cases we prove
a Fadell-Neuwirth type fibration theorem for a class of
Lie groupoids. We show that this is the best possible extension,
in the sense that, for the class of Lie
groupoids corresponding to global quotient orbifolds with nonempty
singular set, the fibration theorems do not hold.
Secondly, we prove a short exact sequence of
fundamental groups (called {\it pure orbifold braid groups}) of one
of the configuration Lie groupoids of the Lie groupoid corresponding
to the punctured complex plane
with cone points.
As consequences, first we see that the
pure orbifold braid groups have poly-virtually free structure, which
generalizes the classical braid group case. We also provide an explicit set of
generators of the pure orbifold braid groups. Secondly, we prove
that a class of affine and finite complex Artin groups are virtually poly-free,
which partially answers the question if all Artin
groups are virtually poly-free ([[3], Question 2]). Finally, combining this poly-
virtually free structure and a recent result ([4]), we deduce the Farrell-Jones
isomorphism conjecture for the above class of orbifold braid groups. This also
implies the conjecture for the case of the Artin group of type D_n tilde, which was
left open in [24].
Click pdf for the full article.
Click jpg1, jpg2 to see Figure 10.
Click movie to see how Figure 10 is obtained.
Click movie to see how Figure 11 (i < k) is obtained. The same trick can be used to see all the others figures in the proof of Proposition 4.6.
Click jpg to see Figure 11 (i < k) final picture.
