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

TL;DR

This paper generalizes the construction of essential surfaces in 3-manifolds from $SL_2$-character varieties to higher dimensions, creating branched surfaces from ideal points of $SL_n$-character varieties for $n \\geq 3$ and linking them to group presentations.

Contribution

It introduces a new method to construct branched surfaces from $SL_n$-character varieties, extending the classical $SL_2$ approach to higher dimensions.

Findings

Constructs essential tribranched surfaces from $SL_n$-character varieties.

Shows these surfaces induce nontrivial group presentations.

Establishes a link between character varieties and 3-manifold splittings.

Abstract

In 1983 Culler and Shalen established a way to construct essential surfaces in a 3-manifold from ideal points of the $SL_2$-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 $SL_n$-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.