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# Mathematical Research Letters

## Volume 25 (2018)

### Number 3

### Representation varieties detect essential surfaces

Pages: 803 – 817

DOI: https://dx.doi.org/10.4310/MRL.2018.v25.n3.a4

#### Authors

#### 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 $\mathrm{SL}_n$-character variety. There exists an essential surface in some $3$-manifold known to be not detected in the classical $\mathrm{SL}_2$-theory. We prove that every connected essential surface in a $3$-manifold is given by an ideal point of a rational curve in the $\mathrm{SL}_n$-character variety for some $n$.

#### Keywords

$3$-manifold, essential surface, character variety, Bruhat–Tits building, separable subgroup

#### 2010 Mathematics Subject Classification

Primary 57N10. Secondary 20E42, 57M05.

Received 13 August 2016

Accepted 25 July 2017

Published 3 August 2018