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I'll try to flesh out the first comment to give an answer: for all $n\ge 4$ there is a representation $\rho:A_{n+1}\to \rm SL_n(\bf Z)$ whose image normally generates $\rm SL_n(\bf Z)$ (because it inj …

For the new question an answer is given by arithmetic Fuchsian groups. For example it is well-known that the reflection group associated with the regular right-angled pentagon in $\mathbb H^2$ contain …

Here is, at Moishe Kohan's request, an optimal answer to the original question (which asked for a representation over any number field; as i noted in the comments there exists plenty of surface group …

This is well-known, and you don't need semisimplicity for this to hold. You can prove it by considering congruence subgroups of $H(\mathcal O_k)$ as follows. Let $\mathfrak n$ be an ideal of $\mathcal …

In this answer i will follow Yves' comment and add references. If $U = \mathbf{U}(R)$ with $\mathbf U$ an algebraic unipotent $\mathbb Q$-group then the two following facts hold : If $\Lambda \le U …

In general approximation or continuity results for various invariants and functions might be used to extend results from the class $C$ to the class of $C$-approximable groups. A concrete example is …

Setting Let $k$ be a real quadratic field, $\mathbb Z_k$ its ring of integers. Let $n$ be an even integer $A$ a symmetric $n$-by-$n$ matrix with coefficients in $\mathbb Z_k$. Let $L$ be the lattice $ …