An invitation to Culler-Shalen theory in arbitrary characteristic

2024/11/11 by Grace S. Garden, Stephan Tillmann, Garden, Grace S. +1 · 2 citations
Mathematics · #57M25 #57N10 #Advanced Algebra and Geometry #FOS: Mathematics #Geometric Topology (math.GT)

paper · pdf · doi:10.48550/arxiv.2411.06859

openalex publication_date 2024/11/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In the seminal work of Culler and Shalen from 1983, essential surfaces in 3-manifolds are associated to ideal points of their SL₂(ℂ)-character varieties, and connections between the algebraic geometry of the character variety and the topology of the 3-manifold are established via group actions on trees. Here, we lay a general foundation for this theory in arbitrary characteristic by using the same approach instead over an arbitrary algebraically closed field. Examples include a change in the A-polynomial in characteristic 2, a closed Haken hyperbolic 3-manifold with no detected essential surface, and a closed Haken hyperbolic 3-manifold with an essential surface only detected in characteristic 2.

Cited by

• On the non-Archimedean Hitchin map for SL₂(F)
• Finite covers and strict boundary slopes of cusped hyperbolic 3-manifolds

Related

• Representation varieties detect essential surfaces
• Ideal points of character varieties, algebraic non-integral representations, and undetected closed essential surfaces in 3-manifolds
• Modules, fields of definition, and the Culler-Shalen norm
• Character varieties of higher dimensional representations and splittings of 3-manifolds
• Symmetries and Detection of Surfaces by the Character Variety