A classical work of M. Culler and P. B. Shalen [Ann. Math. (2) 117, 109–146 (1983; Zbl 0529.57005)] provides a method to construct an essential surface in a \(3\)-manifold from an ideal point in a rational curve in its \(\mathrm{SL}_2(\mathbb{C})\) character variety. However it is known that in general, not every essential surface arises that way. T. Hara and T. Kitayama extended Culler-Shalen’s theory for \(\mathrm{SL}_n(\mathbb{C})\) character varieties [“Character varieties of higher dimensional representations and splittings of 3-manifolds”, Preprint, arXiv:1410.4295] for arbitrary \(n\). Given an ideal point in a curve of the \(\mathrm{SL}_n(\mathbb{C})\) character variety of a \(3\)-manifold, they constructed an essential tri-branched surface which might be an essential surface.
The main result of the present paper is that every essential surface of a given \(3\)-manifold can be constructed from such an ideal point in a curve of its \(\mathrm{SL}_n(\mathbb{C})\) character variety for some positive integer \(n\). The proof is based on a theorem of P. Przytycki and D. T. Wise [Compos. Math. 150, No. 9, 1623–1630 (2014; Zbl 1305.57039)] which states that the fundamental group of an essential surface is separable in the fundamental group of the \(3\)-manifold.