# Geometric conditions for matrix domination in two dimensions

## TL;DR

This paper establishes geometric criteria for matrix domination in two dimensions, linking eigenvector and trace properties with hyperbolic geometry to enable explicit computation and construction.

## Contribution

It introduces necessary and sufficient geometric conditions for domination in the special linear group, including an explicit algorithm for constructing dominated sets with given eigenvectors.

## Key findings

- Derived explicit geometric conditions for matrix domination
- Provided an algorithm for constructing dominated sets with prescribed eigenvectors
- Connected matrix domination to two-dimensional hyperbolic geometry

## Abstract

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.