arXiv subjects

Charles Katerba

Publications and source records attributed to Charles Katerba.

2 recordsLinked to original sources

arXiv2018

Ideal points of character varieties, algebraic non-integral representations, and undetected closed essential surfaces in 3-manifolds

Closed essential surfaces in a three-manifold can be detected by ideal points of the character variety or by algebraic non-integral representations. We give examples of closed essential surfaces not detected in either of these ways. For ideal points, we use Chesebro's module-theoretic interpretation of Culler-Shalen theory. As a corollary, we construct an infinite family of closed hyperbolic Haken 3-manifolds with no algebraic non-integral representations into PSL(2, C), resolving a question of Shanuel and Zhang.

math.GT↗

arXiv2018

Modules, fields of definition, and the Culler-Shalen norm

Culler-Shalen theory uses the algebraic geometry of the SL(2,C)-character variety of a 3-manifold to construct essential surfaces in the manifold. There are module structures associated to the coordinate rings of the irreducible components of character varieties that are intimately related to essential surface construction. When these modules are finitely generated, we derive a formula for their rank that incorporates the irreducible component's field of definition and the Culler-Shalen norm.

math.GT↗