Punctured JSJ tori and tautological extensions of Azumaya algebras

Wang, Yi

doi:10.2140/agt.2026.26.2049

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ISSN (electronic): 1472-2739

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Punctured JSJ tori and tautological extensions of Azumaya algebras

Yi Wang

Algebraic & Geometric Topology 26 (2026) 2049–2077

DOI: 10.2140/agt.2026.26.2049

Abstract

The ${SL}_{2} (ℂ)$ -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 derived by 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.

Keywords

character variety, detected surface, Azumaya algebra

Mathematical Subject Classification

Primary: 57K31

Secondary: 16H05

References

Publication

Received: 29 January 2024

Revised: 7 April 2025

Accepted: 30 May 2025

Published: 22 June 2026

Authors

Yi Wang
Department of Mathematics University of Illinois Urbana-Champaign Urbana, IL United States

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Volume 26, issue 6 (2026)