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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