TITLE: Which group do two generic $2\times 2$ triangular matrices generate?
QUESTION [8 upvotes]: 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?

REPLY [16 votes]: 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.)