Computational Methods for Investigating Higher Teichmüller Spaces

This paper deals with the discrete subgroup problem, solvable through the study of higher Teichmüller theory. The discrete subgroup problem is as follows: given a Lie group (a group with continuity) such as R, how do you find discrete subgroups, such as Z, particularly those of a type similar to (isomorphic to) a certain group? This paper details how, using theorems pertaining to Higher Teichmüller theory graphing the eigenvalue gaps of a matrix group can be used to figure out if it is a discrete subgroup of a Lie Group. In it, I find two subsets of results: one is working towards an attempt to validate recently found results about a Lie group, and the other is a result about the method itself (specifically, projective convex bending) and how it can be used empirically.