Torsion subgroups of quasi-abelianized braid groups

Abstract

This article deals with braid groups of complex reflection groups as introduced in [2]. We present results in two directions. First, we extend the works of Gonçalves, Guaschi, Ocampo [5] and Marin [7] on finite subgroups of the quotients of generalized braid groups by the derived subgroup of their pure braid group. We get explicit criteria for subgroups of the (complex) reflection group to lift to subgroups of this quotient. In the specific case of the classical braid group, this enables us to describe all its finite subgroups: we show that every odd-order finite group can be embedded in it, when the number of strands goes to infinity. We also determine a complete list of the irreducible reflection groups for which this quotient is a Bieberbach group. In the second part, we describe the abelianization of the inverse image of a subgroup of a reflection group in its generalized braid group generalizing results of the first author [1]. In particular, these abelianized groups are not necessarily torsion-free.