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Let $G_{\alpha,\beta,\gamma}$ be a hyperbolic triangle group. Then $G$ has a following presentation. $G=\langle a, b; a^{\alpha}, b^{\beta}, c^{\gamma}, abc\rangle, \alpha, \beta, \gamma \geq 2$. Suppose that $G_{\alpha,\beta,\gamma}$ is a finite index subgroup of some other hyperbolic triangle group $G_{\alpha_{1},\beta_{1},\gamma_{1}}$.Suppose that $\Gamma$ is a subgroup of $SL(3,\mathbb{R})$ isomorphic to $G_{\alpha,\beta,\gamma}$. Then is it true that there exists a group $\Gamma_{1}$ in $SL(3,\mathbb{R})$ which is isomorphic to $G_{\alpha_{1},\beta_{1},\gamma_{1}}$ and contains $\Gamma$ as a finite index subgroup? Is there any counterexample or some condition that this happens? What happens in $SL(2, \mathbb{R})$?

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If you have the extra conditions on embedding of $\Gamma$ in $SL(3,R)$, namely, acting properly on a convex domain in $RP^2$, then the answer is yes. Otherwise, not enough is known about representations of triangle groups to $SL(3,R)$. For counter-examples: Do you really need a representation to $SL(3,R)$ to be injective? Injectivity is hard to verify, maybe a weaker form of injectivity would be enough for you. There are potential (nondiscrete of course) counter-examples in the case of $SL(2,R)$: There is no obvious kernel, but I am not sure if the representations are injective. – Misha Aug 14 '12 at 4:13
For the case when $\Gamma$ acts properly on a convex domain in $RP^{2}$, is the same statement true for every such $\Gamma$ and $\Gamma_{1}$(when $\Gamma$ and $\Gamma_{1}$ are not necessarily triangle groups)? Is there a reference for the proof? Thanks. – kchoi Aug 14 '12 at 4:32
The statement fails for non-triangle groups and their representations to $SL(2,R)$. Say, a generic closed hyperbolic surface $S$ of genus $\ge 3$, does not isometrically cover any hyperbolic orbifold (except for $S$ itself), and group embeddings as in your question, correspond to such coverings. The statement about coverings is a simple corollary of calculation of dimensions of Teichmuller spaces of hyperbolic orbifolds. The argument for triangle groups is not in the literature and I would have to write it up. The key is that spaces of convex-projective structures on ... – Misha Aug 14 '12 at 4:55
...orbifolds of the type $S^2(\alpha,\beta,\gamma)$ are all diffeomorphic to $R^2$ as long as $\alpha,\beta,\gamma>2$. This fact is due to Weil-Hitchin-Goldman-Choi. (The dimension count here fails if one of your generators is an involution, so there will be counter-examples in this case.) The last ingredient is that one space of projective structures is properly embedded in the other, so they have to be the same. This is again essentially due to Hitchin-Goldman-Choi. – Misha Aug 14 '12 at 5:09
The counter-examples will be of the type $G_{n,n,n}< G_{2,3,2n}$, $n\ge 4$. There will be discrete embeddings of $G_{n,n,n}$ to $SL(3,R)$ which do not extend to $G_{2,3,2n}$. Same for $G_{n,n,m}<G_{2,2m,n}$. But these are the only counter-examples in the context of groups acting properly on convex domains. – Misha Aug 14 '12 at 5:27

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