All Questions

Let $G\subset \operatorname{GL}_n$ be a linear algebraic group over $\mathbb{Q}$ and let $\Gamma\subset G\cap \operatorname{GL}_n(\mathbb{Z})$ be an arithmeric subgroup without torsion. Using a result ...

Let $A,B \in \mathrm{SL}_3(\mathbb{Z})$. Set $$S = \langle A \rangle \cdot \langle B \rangle = \{A^mB^n : m,n \in \mathbb{Z}\}.$$ Is $S$ closed in the profinite topology on $\mathrm{SL}_3(\mathbb{...

If so, is there a way to conclude this from Malcev's theorem? In general, what is known about virtually torsion freeness of non-finitely generated linear groups?

Before stating my question, let me recall the Congruence Subgroup Property/Problem: Given simply connected absolutely and almost simple algebraic group $G$ with fixed realization as a matrix group one ...