Punctured JSJ tori and tautological extensions of Azumaya algebras

TL;DR

This paper develops a topological criterion using Culler-Shalen theory and JSJ decompositions to extend Azumaya algebras over ideal points of character varieties, leading to potentially finer arithmetic invariants for hyperbolic 3-manifolds.

Contribution

It introduces an explicit topological criterion for extending Azumaya algebras over ideal points, enhancing the understanding of arithmetic invariants in hyperbolic 3-manifold topology.

Findings

Criteria for extending Azumaya algebras over ideal points

Examples of refined invariants in specific cases

Connection between Seifert surfaces and ideal points

Abstract

The $SL_2(\mathbb{C})$ character variety $X(M)$ has emerged as an important tool in studying the topology of hyperbolic 3-manifolds. Chinburg-Reid-Stover constructed arithmetic invariants stemming from a canonical Azumaya algebra over the normalization of an irreducible component of $X(M)$ containing a lift of the holonomy representation of $M$. We provide an explicit topological criterion for extending the canonical Azumaya algebra over an ideal point, potentially leading to finer arithmetic invariants than those of Chinburg-Reid-Stover. This topological criterion involves Culler-Shalen theory and, in some cases, JSJ decompositions of toroidal Dehn fillings of knot complements in the three-sphere. Inspired by the work of Paoluzzi-Porti and Tillmann, we provide examples of several cases where these refined invariants exist. Along the way, we show that certain families of Seifert surfaces in hyperbolic knot complements can be associated to ideal points of character varieties.