Are finitely generated subgroups of $PSL_n(\mathbb{R})$ linear?

Question

Can finitely generated subgroups of $PSL_2(\mathbb{R})$ necessarily be embedded into $GL_n(\mathbb{R})$. In particular, I would like a linear representation for the fundamental group of a closed orientable surface of genus greater than or equal to 2 (which we can naturally embed in $PSL_2(\mathbb{R})$ by viewing the surface as a quotient of the hyperbolic plane by a certain group of isometries).

Answer 1

$PSL_2(\mathbb{R})$ itself is a linear group: there is a smooth representation $$\rho : PSL_2(\mathbb{R}) \longrightarrow GL(\mathbb{R}^3)$$ defined by $$\rho(A)(x,y,z) = (x',y',z')$$ where $$\begin{pmatrix} x' & y' \\ y' & z' \end{pmatrix} = A \begin{pmatrix} x & y \\ y & z \end{pmatrix} A^T.$$ As $PSL_2(\mathbb{R})$ is simple this map must be an embedding.