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.



Can Galois conjugates of lattices in SL(2,R) be discrete?
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](http://en.wikipedia.org/wiki/Supersingular_prime_%28moonshine_theory%29) $X\_0^+(N)$ 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](http://en.wikipedia.org/wiki/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](http://en.wikipedia.org/wiki/Markoff_equation). In [Proposition 4.11](http://homepages.warwick.ac.uk/~masgak/abstracts/qfg.html), 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. 


