Let $G$ be a semisimple algebraic $\mathbb{Q}$-group and $\Gamma$ an arithmetic subgroup of $G$. In particular $\Gamma$ is finitely generated.
Denote by $\Gamma^{u}$ the set of unipotent elements in $\Gamma$. My main question is the following.
Question 1.: Is the group $\langle \Gamma^{u} \rangle$ generated by $\Gamma^{u}$ also finitely generated?
If the simple factors of $G$ are not of rank $1$, then for example the description of the normal subgroups by Margulis yields a positive answer. However, I am looking for a general answer which preferably doesn't use the latter.
Next, define also the group $N$ generated by the elements of $\Gamma^{u}$ lying in one of the factors of $G$.
Question 2: Is there any general results on the quotient group $\langle \Gamma^{u} \rangle/ N$?
If one defines all the above over $G$ instead of $\Gamma$ then such quotient was investigated and for example is always a finite group.
Assuming the following setup would in fact already suffice for me:
- $G \cong \prod_{i=1}^q GL_{n_i}(D_i)$ with $D_i$ a finite dimensional division $\mathbb{Q}$-algebra
- $\Gamma$ a subgroup of finite index in $\prod_{i=1}^q GL_{n_i}(\mathcal{O}_i)$ with $\mathcal{O}_i$ a ($\mathbb{Z}$-)order in $D_i$
