We can get arithmetic lattices isomorphic to free groups in $\mathrm{SL}_2\mathbb{R}$, so in general we can’t expect homomorphisms of lattices into semisimple Lie groups to say much about $\mathrm{SL}_2\mathbb{R}$, even if we require the image to be discrete. In particular, unlike in higher-rank cases, such homomorphisms won’t necessarily extend to homomorphisms of $\mathrm{SL}_2\mathbb{R}$.

However, if $\Gamma<\mathrm{SL}_2\mathbb{R}$ is cocompact and we have an discrete embedding (by which I mean a map which is both a homeomorphism and an isomorphism onto its image) of $\Gamma$ into some semisimple Lie group $G$, will we not necessarily get a copy of $\mathrm{SL}_2\mathbb{R}$ in $G$ containing (the image of) $\Gamma$?

By the considerations above, the image of $\Gamma$ in such a copy of $\mathrm{SL}_2\mathbb{R}$ might not look like it originally did (meaning the embedding of $\Gamma$ hasn’t necessarily extended to all of $\mathrm{SL}_2\mathbb{R}$), but part of me imagines that the image of $\Gamma$ should still determine something. Is this naïve?