An invitation to Culler-Shalen theory in arbitrary characteristic
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866929626560856064 |
|---|---|
| author | Garden, Grace S. Tillmann, Stephan |
| author_facet | Garden, Grace S. Tillmann, Stephan |
| contents | In the seminal work of Culler and Shalen from 1983, essential surfaces in 3-manifolds are associated to ideal points of their $\text{SL}_2(\mathbb{C})$-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_06859 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An invitation to Culler-Shalen theory in arbitrary characteristic Garden, Grace S. Tillmann, Stephan Geometric Topology 57M25, 57N10 In the seminal work of Culler and Shalen from 1983, essential surfaces in 3-manifolds are associated to ideal points of their $\text{SL}_2(\mathbb{C})$-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. |
| title | An invitation to Culler-Shalen theory in arbitrary characteristic |
| topic | Geometric Topology 57M25, 57N10 |
| url | https://arxiv.org/abs/2411.06859 |