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Let $H$ be an anisotropic $\mathbb{Q}$-form of $SL_2$, with $H(\mathbb{R}) = SL_2(\mathbb{R})$.

Are there any results or conjecture regarding the existence, or non-existence of intermediate subgroups $\Gamma$ between $H(\mathbb{Z})$ and $H(\mathbb{Z}[\sqrt{2}])$ such that $\Gamma$ is of infinite index in $H(\mathbb{Z}[\sqrt{2}])$ and $H(\mathbb{Z})$ is of infinite index in $\Gamma$ ?

When $H$ is isotropic, there are no such groups; this was proven by Venkataramana in "On some rigid subgroups of semisimple Lie groups." Israel J. Math. 89 (1995), no. 1-3, 227–236, doi: 10.1007/BF02808202.

Let $H$ be an anisotropic $\mathbb{Q}$-form of $SL_2$, with $H(\mathbb{R}) = SL_2(\mathbb{R})$.

Are there any results or conjecture regarding the existence, or non-existence of intermediate subgroups $\Gamma$ between $H(\mathbb{Z})$ and $H(\mathbb{Z}[\sqrt{2}])$ such that $\Gamma$ is of infinite index in $H(\mathbb{Z}[\sqrt{2}])$ and $H(\mathbb{Z})$ is of infinite index in $\Gamma$ ?

When $H$ is isotropic, there are no such groups; this was proven by Venkataramana in "On some rigid subgroups of semisimple Lie groups." Israel J. Math. 89 (1995), no. 1-3, 227–236, doi: 10.1007/BF02808202. See Proposition 2.1.

Let $H$ be an anisotropic $\mathbb{Q}$-form of $SL_2$, with $H(\mathbb{R}) = SL_2(\mathbb{R})$.

Are there any results or conjecture regarding the existence, or non-existence of intermediate subgroups $\Gamma$ between $H(\mathbb{Z})$ and $H(\mathbb{Z}[\sqrt{2}])$, such that $\Gamma$ is of infinite index in $H(\mathbb{Z}[\sqrt{2}])$ and $H(\mathbb{Z})$ is of infinite index in $\Gamma$ ?

When $H$ is isotropic, there are no such groups; this was proven by Venkataramana in "On some rigid subgroups of semisimple Lie groups." Israel J. Math. 89 (1995), no. 1-3, 227–236, doi: 10.1007/BF02808202.

Let $H$ be an anisotropic $\mathbb{Q}$-form of $SL_2$, with $H(\mathbb{R}) = SL_2(\mathbb{R})$.

Are there any results or conjecture regarding the existence, or non-existence of intermediate subgroups $\Gamma$ between $H(\mathbb{Z})$ and $H(\mathbb{Z}[\sqrt{2}])$ such that $\Gamma$ is of infinite index in $H(\mathbb{Z}[\sqrt{2}])$ and $H(\mathbb{Z})$ is of infinite index in $\Gamma$ ?

When $H$ is isotropic, there are no such groups; this was proven by Venkataramana in "On some rigid subgroups of semisimple Lie groups." Israel J. Math. 89 (1995), no. 1-3, 227–236, doi: 10.1007/BF02808202.

Let $H$ be an anisotropic $\mathbb{Q}$-form of $SL_2$, with $H(\mathbb{R}) = SL_2(\mathbb{R})$.

Are there any results or conjecture regarding the existence, or non-existence of intermediate subgroups $\Gamma$ between $H(\mathbb{Z})$ and $H(\mathbb{Z}[\sqrt{2}])$, such that $\Gamma$ is of infinite index in $H(\mathbb{Z}[\sqrt{2}])$ and $H(\mathbb{Z})$ is of infinite index in $\Gamma$ ?

When $H$ is isotropic, there are no such groups; this was proven by Venkataramana in "On some rigid subgroups of semisimple Lie groups." Israel J. Math. 89 (1995), no. 1-3, 227–236.

Let $H$ be an anisotropic $\mathbb{Q}$-form of $SL_2$, with $H(\mathbb{R}) = SL_2(\mathbb{R})$.

Are there any results or conjecture regarding the existence, or non-existence of intermediate subgroups $\Gamma$ between $H(\mathbb{Z})$ and $H(\mathbb{Z}[\sqrt{2}])$, such that $\Gamma$ is of infinite index in $H(\mathbb{Z}[\sqrt{2}])$ and $H(\mathbb{Z})$ is of infinite index in $\Gamma$ ?

When $H$ is isotropic, there are no such groups; this was proven by Venkataramana in "On some rigid subgroups of semisimple Lie groups." Israel J. Math. 89 (1995), no. 1-3, 227–236, doi: 10.1007/BF02808202.