UC Santa Barbara

This thesis presents a number of results surrounding the arithmetic properties of the Hitchin component, a space of representations of the fundamental group of a closed surface into split real forms of simple Lie groups such as $\rm{SL}(n,\mathbf{R})$. The first main result is when the existence of a Hitchin representation defined over certain prescribed subrings of $\R$ may be deformed to a Zariski-dense one defined over the same ring. The second, produced in joint work with Jacques Audibert, provides a topological characterization for the collection of Hitchin representations defined over $\mathbf{Q}$. Central to establishing these results is further developing the arithmetic nature of bending deformations on the Hitchin component. This further develops a perspective first taken in work of Long and Thistlethwaite who studied these deformations in an arithmetic context in order to produce examples of thin surface subgroups of $\rm{SL}(2k+1,\mathbf{Z})$.