Character varieties of higher dimensional representations and splittings of 3-manifolds

Abstract

In 1983 Culler and Shalen established a way to construct essential surfaces in a 3-manifold from ideal points of the \(\mathrm {SL}_2\)-character variety associated to the 3-manifold group. We present in this article an analogous construction of certain kinds of branched surfaces (which we call essential tribranched surfaces) from ideal points of the \(\mathrm {SL}_n\)-character variety for a natural number n greater than or equal to 3. Further we verify that such a branched surface induces a nontrivial presentation of the 3-manifold group in terms of the fundamental group of a certain 2-dimensional complex of groups.