Coxeter groups are a special class of groups generated by involutions. They play important roles in the various areas of mathematics. This survey particularly focuses on how one use Coxeter groups to construct interesting examples of discrete subgroups of Lie group.
We extend the notion of Hitchin component from surface groups to orbifold groups and prove that this gives new examples of higher Teichm{u}ller spaces. We show that the Hitchin component of an orbifold group is homeomorphic to an open ball and we com
pute its dimension explicitly. We then give applications to the study of the pressure metric, cyclic Higgs bundles, and the deformation theory of real projective structures on $3$-manifolds.
