For $\textrm{ SL}(n,\mathbb{ R})$ ($n\geq3$), $\textrm{ SO}(n+1,n)$ ($n\geq2$), $\textrm{ Sp}(2n,\mathbb{ R})$ ($n\geq2$) and for the adjoint real split form of the exceptional group $\textrm{ G}_2$, we exhibit non-uniform lattices in which we construct thin Hitchin representations by arithmetic methods. These representations give infinitely many orbits under the action of the mapping class group (except maybe for $\textrm{ G}_2$). In particular, we show that when $p\neq2$ is prime every non-uniform lattice of $\mathrm{ SL}(p,\mathbb{ R})$ contains thin Hitchin representations.
Categories: math.GT math.GR math.NT