Torsion subgroups of quasi-abelianized braid groups

TL;DR

This paper characterizes finite subgroups of certain braid group quotients, providing explicit criteria for lifting subgroups and describing all finite subgroups of the classical braid group, especially for odd orders as the number of strands grows.

Contribution

It offers explicit criteria for subgroup lifting in braid group quotients and classifies finite subgroups of the classical braid group, including all odd-order groups for large strand numbers.

Findings

Every odd-order finite group embeds in the classical braid group as strands increase.

Complete classification of reflection groups with Bieberbach group quotients.

Explicit subgroup lifting criteria for complex reflection groups.

Abstract

This article extends the works of Gon\c{c}alves, Guaschi, Ocampo [GGO] and Marin [MAR2] 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.