Representation varieties detect essential surfaces

TL;DR

This paper extends the theory of character varieties to higher dimensions, showing that all essential surfaces in 3-manifolds can be detected via ideal points in $ ext{SL}_n$-character varieties, broadening the scope of surface detection methods.

Contribution

It proves that every essential surface in a 3-manifold corresponds to an ideal point in some $ ext{SL}_n$-character variety, generalizing classical detection techniques.

Findings

Every essential surface is detected by an ideal point in some $ ext{SL}_n$-character variety.

Extension of Culler-Shalen theory to higher rank groups.

Demonstrates the existence of essential surfaces not detected by classical $ ext{SL}_2$-theory.

Abstract

Extending Culler-Shalen theory, Hara and the second author presented a way to construct certain kinds of branched surfaces in a $3$-manifold from an ideal point of a curve in the $\operatorname{SL}_n$-character variety. There exists an essential surface in some $3$-manifold known to be not detected in the classical $\operatorname{SL}_2$-theory. We prove that every connected essential surface in a $3$-manifold is given by an ideal point of a rational curve in the $\operatorname{SL}_n$-character variety for some $n$.