The G_2-Hitchin Component of Triangle Groups: Dimension and Integer Points

The image of $\PSL(2,\reals)$ under the irreducible representation into $\PSL(7,\reals)$ is contained in the split real form $G_{2}^{4,3}$ of the exceptional Lie group $G_{2}$. This irreducible representation therefore gives a representation $\rho$ of a hyperbolic triangle group $\Gamma(p,q,r)$ into $G_{2}^{4,3}$, and the \textit{Hitchin component} of the representation variety $\Hom(\Gamma(p,q,r),G_{2}^{4,3})$ is the component of $\Hom(\Gamma(p,q,r),G_{2}^{4,3})$ containing $\rho$.

This thesis is in two parts: (i) we give a simple, elementary proof of a formula for the dimension of this Hitchin component, this formula having been obtained earlier in [Alessandrini et al.], \citep{Alessandrini2023}, as part of a wider investigation using Higgs bundle techniques, and (ii) we prove the existence of an infinite sequence of integer points on the $G_{2}$-Hitchin component of the (2,4,6)-triangle group.

One reason for studying hyperbolic triangle groups is that they contain surface groups as subgroups of finite index. Integer representations in Hitchin components then often provide examples of surface groups represented as elusive \textit{thin matrix groups}, see [Sarnak] \citep{Sarnak2013}, [Long and Reid] \citep{LongReid2013}, and [Kontorovich et al.] \citep{Kontorovich2019}.