Geometric conditions for matrix domination in two dimensions

In this article we prove a necessary and a sufficient condition for a finite subset of the special linear group to be dominated. These conditions are purely geometric in nature, as they only involve the trace and the eigenvectors of the matrices, and can be computed explicitly. Our sufficient condition, in particular, provides a simple algorithm for constructing a dominated set with prescribed eigenvectors. The techniques involved in our proofs take advantage of the interaction between dominated sets and two-dimensional hyperbolic geometry.