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 \(n = 3\).