Extending Azumaya algebras associated to arithmetic 2-bridge links

TL;DR

This paper explores the algebraic structures related to hyperbolic 3-manifolds and links, demonstrating that certain quaternion algebras associated with these structures do not extend to Azumaya algebras over canonical surfaces.

Contribution

It generalizes previous results on character varieties to hyperbolic link complements and shows the non-extension of associated quaternion algebras to Azumaya algebras.

Findings

Canonical quaternion algebra does not extend to an Azumaya algebra over the canonical surfaces.

Generalizes algebraic and arithmetic geometry tools to hyperbolic link complements.

Provides new insights into the algebraic structures of hyperbolic 3-manifolds.

Abstract

Let {\Gamma} be a finitely generated group and consider the set of all characters of representations of {\Gamma} into SL2(C). This set, denoted by X({\Gamma}), admits an algebraic structure and is called the character variety of {\Gamma}. When {\Gamma} is the fundamental group of a hyperbolic 3-manifold M, X({\Gamma}) turns out to be a powerful tool in the study of the geometry and topology of M. Chinburg-Reid-Stover have borrowed tools from algebraic and arithmetic geometry to understand algebraic and number-theoretic properties of the canonical curves of X({\Gamma}). In this paper, we will partly generalize their results to certain hyperbolic link complements, and prove that the associated canonical quaternion algebra will not extend to an Azumaya algebra over the canonical surfaces.