The Asymptotics of Representations for Cyclic Opers

Given a Riemann surface $X = (\Sigma, J)$ we find an expression for the dominant term for the asymptotics of the holonomy of opers over that Riemann surface corresponding to rays in the Hitchin base of the form $(0,0,\cdots,t\omega_n)$. Moreover, we find an associated equivariant map from the universal cover $(\tilde{\Sigma},\tilde{J})$ to the symmetric space SL$_n(\mathbb{C}) / \mbox{SU}(n)$ and show that limits of these maps tend to a sub-building in the asymptotic cone. That sub-building is explicitly constructed from the local data of $\omega_n$.