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Let $G$ be a simply connected complex Chevalley group and let $T$ be some maximal torus in $G$ with $\dim T\geq 2$.

From "A note on generators for arithmetic subgroups of algebraic groups" by Raghunathan, we have that for every ideal $I\lhd\mathbb Z$, $$\Gamma(I):=\langle U_\phi(I)|\phi\in\Phi \rangle$$ is of finite index in $G(\mathbb Z)$.

Is it also true for every ideal $I\lhd\mathbb Z[\alpha]$ that $Γ(I)$ of finite index in $G(\mathbb Z[\alpha])$?

Specifically, I'm looking at $G=SL_n$

Where:

  • $\Phi$ denote the root system of $G$ with respect to $T$;
  • $U_\phi$ is the root subgroup corresponding to $\phi\in\Phi$;
  • $\alpha$ is an element that is integral over $\mathbb Z$;
  • if $I$ is an ideal of the ring $A$ and $H\leq G$ we mark $H(I):=\{x\in H(A)|x\equiv1(\mathrm{mod}\,I)\}$ where $H(A):=H\cap \mathrm{GL}_n(A)$.

Let $G$ be a simply connected complex Chevalley group and let $T$ be some maximal torus in $G$ with $\dim T\geq 2$.

From "A note on generators for arithmetic subgroups of algebraic groups" by Raghunathan, we have that for every ideal $I\lhd\mathbb Z$, $$\Gamma(I):=\langle U_\phi(I)|\phi\in\Phi \rangle$$ is of finite index in $G(\mathbb Z)$.

Is it also true for every ideal $I\lhd\mathbb Z[\alpha]$ that $Γ(I)$ of finite index in $G(\mathbb Z[\alpha])$?

Where:

  • $\Phi$ denote the root system of $G$ with respect to $T$;
  • $U_\phi$ is the root subgroup corresponding to $\phi\in\Phi$;
  • $\alpha$ is an element that is integral over $\mathbb Z$;
  • if $I$ is an ideal of the ring $A$ and $H\leq G$ we mark $H(I):=\{x\in H(A)|x\equiv1(\mathrm{mod}\,I)\}$ where $H(A):=H\cap \mathrm{GL}_n(A)$.

Let $G$ be a simply connected complex Chevalley group and let $T$ be some maximal torus in $G$ with $\dim T\geq 2$.

From "A note on generators for arithmetic subgroups of algebraic groups" by Raghunathan, we have that for every ideal $I\lhd\mathbb Z$, $$\Gamma(I):=\langle U_\phi(I)|\phi\in\Phi \rangle$$ is of finite index in $G(\mathbb Z)$.

Is it also true for every ideal $I\lhd\mathbb Z[\alpha]$ that $Γ(I)$ of finite index in $G(\mathbb Z[\alpha])$?

Specifically, I'm looking at $G=SL_n$

Where:

  • $\Phi$ denote the root system of $G$ with respect to $T$;
  • $U_\phi$ is the root subgroup corresponding to $\phi\in\Phi$;
  • $\alpha$ is an element that is integral over $\mathbb Z$;
  • if $I$ is an ideal of the ring $A$ and $H\leq G$ we mark $H(I):=\{x\in H(A)|x\equiv1(\mathrm{mod}\,I)\}$ where $H(A):=H\cap \mathrm{GL}_n(A)$.
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YCor
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Let $G$ be a simply connected complex Chevalley group and let $T$ be some maximal torus in $G$ with $dimT\geq 2$$\dim T\geq 2$.

From A"A note on generators for arithmetic subgroups of algebraic groups by Raghunathangroups"
weby Raghunathan, we have that for $I\lhd\mathbb Z, Γ(I):=\langle U_\phi(I)|\phi\in\Phi \rangle$ is of finite index inevery ideal $G(\mathbb Z)$$I\lhd\mathbb Z$, is it also true for $I\lhd\mathbb Z[\alpha]$ that $Γ(I)$$$\Gamma(I):=\langle U_\phi(I)|\phi\in\Phi \rangle$$ is of finite index in $G(\mathbb Z[\alpha])$?$G(\mathbb Z)$.

Is it also true for every ideal $I\lhd\mathbb Z[\alpha]$ that $Γ(I)$ of finite index in $G(\mathbb Z[\alpha])$?

Where:
-$\Phi$ denote the root system of $G$ with respect to $T$.
-$U_\phi$ is the root subgroup corresponding to $\phi\in\Phi$.
-$\alpha$ is an integral element of $\mathbb Z$.
-If $I$ is an ideal of the ring $A$ and $H\leq G$ we mark $H(I):=\{x\in H(A)|x\equiv1(mod\,I)\}$ where $H(A):=H\cap GL_n(A)$.

  • $\Phi$ denote the root system of $G$ with respect to $T$;
  • $U_\phi$ is the root subgroup corresponding to $\phi\in\Phi$;
  • $\alpha$ is an element that is integral over $\mathbb Z$;
  • if $I$ is an ideal of the ring $A$ and $H\leq G$ we mark $H(I):=\{x\in H(A)|x\equiv1(\mathrm{mod}\,I)\}$ where $H(A):=H\cap \mathrm{GL}_n(A)$.

Let $G$ be a simply connected complex Chevalley group and let $T$ be some maximal torus in $G$ with $dimT\geq 2$.

From A note on generators for arithmetic subgroups of algebraic groups by Raghunathan
we have that for $I\lhd\mathbb Z, Γ(I):=\langle U_\phi(I)|\phi\in\Phi \rangle$ is of finite index in $G(\mathbb Z)$, is it also true for $I\lhd\mathbb Z[\alpha]$ that $Γ(I)$ of finite index in $G(\mathbb Z[\alpha])$?

Where:
-$\Phi$ denote the root system of $G$ with respect to $T$.
-$U_\phi$ is the root subgroup corresponding to $\phi\in\Phi$.
-$\alpha$ is an integral element of $\mathbb Z$.
-If $I$ is an ideal of the ring $A$ and $H\leq G$ we mark $H(I):=\{x\in H(A)|x\equiv1(mod\,I)\}$ where $H(A):=H\cap GL_n(A)$.

Let $G$ be a simply connected complex Chevalley group and let $T$ be some maximal torus in $G$ with $\dim T\geq 2$.

From "A note on generators for arithmetic subgroups of algebraic groups" by Raghunathan, we have that for every ideal $I\lhd\mathbb Z$, $$\Gamma(I):=\langle U_\phi(I)|\phi\in\Phi \rangle$$ is of finite index in $G(\mathbb Z)$.

Is it also true for every ideal $I\lhd\mathbb Z[\alpha]$ that $Γ(I)$ of finite index in $G(\mathbb Z[\alpha])$?

Where:

  • $\Phi$ denote the root system of $G$ with respect to $T$;
  • $U_\phi$ is the root subgroup corresponding to $\phi\in\Phi$;
  • $\alpha$ is an element that is integral over $\mathbb Z$;
  • if $I$ is an ideal of the ring $A$ and $H\leq G$ we mark $H(I):=\{x\in H(A)|x\equiv1(\mathrm{mod}\,I)\}$ where $H(A):=H\cap \mathrm{GL}_n(A)$.
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Ami
Ami
  • 332
  • 1
  • 8

A group generated by all the root subgroups above ideal of $\mathbb Z[\alpha]$ is of finite index in $G(\mathbb Z[\alpha])$

Let $G$ be a simply connected complex Chevalley group and let $T$ be some maximal torus in $G$ with $dimT\geq 2$.

From A note on generators for arithmetic subgroups of algebraic groups by Raghunathan
we have that for $I\lhd\mathbb Z, Γ(I):=\langle U_\phi(I)|\phi\in\Phi \rangle$ is of finite index in $G(\mathbb Z)$, is it also true for $I\lhd\mathbb Z[\alpha]$ that $Γ(I)$ of finite index in $G(\mathbb Z[\alpha])$?

Where:
-$\Phi$ denote the root system of $G$ with respect to $T$.
-$U_\phi$ is the root subgroup corresponding to $\phi\in\Phi$.
-$\alpha$ is an integral element of $\mathbb Z$.
-If $I$ is an ideal of the ring $A$ and $H\leq G$ we mark $H(I):=\{x\in H(A)|x\equiv1(mod\,I)\}$ where $H(A):=H\cap GL_n(A)$.