On the Gauss-Epple homomorphism of the braid group $B_n$, and generalizations to Artin groups of crystallographic type

TL;DR

This paper introduces and analyzes the Gauss-Epple homomorphism for braid groups, explores its properties, and generalizes it to Artin groups of finite type, revealing new algebraic structures and potential for further generalizations.

Contribution

It defines the Gauss-Epple homomorphism, proves its properties, and extends it to Artin groups of finite type, broadening the understanding of braid group invariants.

Findings

Homomorphism factors through ^n times S_n

Homomorphism's image is an order 2 subgroup

Calculated asymptotic probability of containing a random braid

Abstract

In this paper, we introduce a broad family of group homomorphisms that we name the Gauss-Epple homomorphisms. In the setting of braid groups, the Gauss-Epple invariant was originally defined by Epple based on a note of Gauss as an action of the braid group $B_n$ on the set $\{1, \dots, n\}\times\mathbb{Z}$; we prove that it is well-defined. We consider the associated group homomorphism from $B_n$ to the symmetric group $\text{Sym}(\{1, \dots, n\}\times\mathbb{Z})$. We prove that this homomorphism factors through $\mathbb{Z}^n\rtimes S_n$ (in fact, its image is an order 2 subgroup of the previous group). We also describe the kernel of the homomorphism and calculate the asymptotic probability that it contains a random braid of a given length. Furthermore, we discuss the super-Gauss-Epple homomorphism, a homomorphism which extends the generalization of the Gauss-Epple homomorphism and describe a related 1-cocycle of the symmetric group $S_n$ on the set of antisymmetric $n\times n$ matrices over the integers. We then generalize the super-Gauss-Epple homomorphism and the associated 1-cocycle to Artin groups of finite type. For future work, we suggest studying possible generalizations to complex reflection groups and computing the vector spaces of Gauss-Epple analogues.