Let $\Gamma$ be a lattice in $SL(2,\mathbb{R})$. Suppose that the trace field of $\Gamma$ is a totally real number field of degree $d$. This gives $d$ homomorphisms $\rho_i:\Gamma\to SL(2,\mathbb{R})$ (here $i=1,\ldots, d$), which act as Galois conjugation on traces. Here $\rho_1$ denotes the identity homomorphism.

Question 1: Can any of the $\rho(\Gamma_i)$ be discrete when $i\neq 1$?

Question 2: Can the group $\{(\rho_2(g), \rho_3(g), \ldots, \rho_d(g):g\in \Gamma\}$ ever be discrete in $SL(2,\mathbb{R})^{d-1}$?

A positive answer to the question 1 of course also gives a positive answer to question 2. I am most interested in the case when $\Gamma$ is not cocompact, and I would even be happy to know the answer when $\Gamma$ is a triangle group.