The dimension of the Hitchin component for triangle groups

Abstract

Let $p,q,r$ be positive integers satisfying $1/p + 1/q + 1/r < 1$, and let $\Delta (p,q,r)$ be a geodesic triangle in the hyperbolic plane with angles $\pi /p \,,\,\pi /q \,,\,\pi /r$. Then there exists a tiling of the hyperbolic plane by triangles congruent to $\Delta (p,q,r)$, and we define the triangle group$T(p,q,r)$ to be the group of orientation preserving isometries of this tiling. Representation varieties of closed surface groups into $\mathrm {SL(n,{\mathbb {R}})}$ have been studied extensively by Hitchin and Labourie, and the dimension of a certain distinguished component of the variety was obtained by Hitchin using Higgs bundles. Here we determine the corresponding dimension for representations of triangle groups into $\mathrm {SL(n,{\mathbb {R}})}$, generalising some earlier work of Choi and Goldman in the case $n = 3$.