The Dimension of the Restricted Hitchin Component for Triangle Groups

Given positive integers p, q, r satisfying 1/p + 1/q + 1/r < 1, the hyperbolic triangle group T(p,q,r) is the group of orientation-preserving isometries of a tiling of the hyperbolic plane by triangles congruent to a geodesic triangle with angles π/p, π/q, and π/r. We will examine representations of triangle groups in the Hitchin component, a topologically connected component of the representation variety where representations are always discrete and faithful.We begin by giving a formula for the dimension of a subset of the Hitchin component of an arbitrary hyperbolic triangle T(p, q, r) for general degree n > 2. Depending on whether n is even or odd, we will consider only those Hitchin representations whose images lie in Sp(2m) or SO(m,m + 1), respectively. We call the space of representations satisfying this criterion the restricted Hitchin component.We then provide two new families of representations of the specific triangle group T(3,3,4) into SL(5,R); the image groups of these families are each shown to be Zariski dense in SL(5,R). Further, we consider a restriction to a surface subgroup of finite index in T(3,3,4). For each family, we will demonstrate the existence of a subsequence of representations whose images are pairwise non-conjugate in SL(5,Z) when restricted to a surface subgroup.