Let $A,B$ be two generic (in particular invertible) $2\times 2$ upper-triangular complex matrices. They generate a countable group $G$, the commutator subgroup of $G$ is abelian. Are there other relations in $G$? How is this group called?



Which group do two generic $2\times 2$ triangular matrices generate?
I think it is a old result of Magnus that the free metabelian group on $d$ generators can be embedded as group $T\_2$ of $2\times 2$ matrices over $\mathbf{C}$.


It is easy to deduce that 



> 
> every generic $d$-tuple in $T\_2$ freely generates such a free metabelian group $\Gamma\_d$. 
> 
> 
> 


Here generic means: not in some countable union of proper Zariski closed subsets of $T\_2^d$; hence it's generic both in the topological and the measured sense.


Indeed, let $w(x\_1,\dots,x\_d)$ be a nontrivial element $\Gamma\_d$. Let $F\_w$ be the set of $d$-tuples $g\in T\_2^d$ such that $w(g)=I\_2$. Then $F\_w$ is a proper Zariski closed subset. Hence any $d$-tuple outside $\bigcup\_{w\in\Gamma\_d-\{1\}}F\_w$ works.


(A group is metabelian if its derived subgroup is abelian, or equivalently if and only if any two commutators commute. So it doesn't mean a free metabelian group satisfies no other relation, but it indeed means that all its relations can be derived from this one.)


