Can Galois conjugates of lattices in SL(2,R) be discrete? 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.
 A: You probably want to consider the invariant trace field $k$ instead of trace field $K$. The trace field $K$ is a multi-quadratic extension of $k$, and $Gal(K/k)$ acts trivially on the $PSL(2,\mathbb{R})$ representations of $\Gamma$ (it acts non-trivially on  $\Gamma < SL(2,\mathbb{R})$, but with the same image in $PSL(2,\mathbb{R})$). For example, the lattice associated to the modular curve $X_0^+(N)$ will have trace field $\mathbb{Q}(\sqrt{N})$. This is an extension of the lattice $\Gamma_0(N)$ (which has integral trace) by the Fricke involution. The Galois automorphism sends the involution to its negative in $SL_2(\mathbb{R})$, but has the same image in $PSL(2,\mathbb{R})$, and therefore the same action on $\mathbb{H}^2$. 
The answer to your question is yes for punctured torus groups. This follows from results of Brian Bowditch. A representation of the punctured torus group in $SL(2,\mathbb{R})$ corresponds to a non-zero real solution of the Markoff equation. In Proposition 4.11, it is shown that conversely a non-zero solution gives a discrete fuchsian representation. So if the trace field is totally real, all non-trivial Galois conjugates will be discrete fuchsian groups. 
