Springer categories for regular centralizers in well-generated complex braid groups

TL;DR

This paper studies Springer categories associated with regular centralizers in well-generated complex braid groups, providing new Garside theoretic insights, presentations, and proofs related to their structure and conjugacy properties.

Contribution

It offers a detailed analysis of Springer categories, introduces a Hurwitz-like presentation, and applies a generalized Reidemeister-Schreier method to complex braid groups.

Findings

Conjugacy of braided reflections characterized via Garside structure.

Pure Garside theoretic proof of center properties in complex braid groups.

Explicit presentations of the complex braid group B(G31).

Abstract

In his proof of the K(pi,1) conjecture for complex reflection arrangements, Bessis defined Garside categories suitable for studying braid groups of centralizers of Springer regular elements in well-generated complex reflection groups. We provide a detailed study of these categories, which we call Springer categories. We describe in particular the conjugacy of braided reflections of regular centralizer in terms of the Garside structure of the associated Springer category. In so doing we obtain a pure Garside theoretic proof of a theorem of Digne, Marin and Michel on the center of finite index subgroups in complex braid groups in the case of a regular centralizer in a well-generated group. We also provide a "Hurwitz-like" presentation of Springer categories. To this aim we provide additional insights on noncrossing partitions in the infinite series. Lastly, we use this "Hurwitz-like" presentation, along with a generalized Reidemeister-Schreier method we introduce for groupoids, to deduce nice presentations of the complex braid group B(G31).