On the generic triangle group and the free metabelian group of rank 2

We introduce the concept of a generic Euclidean triangle $ au$ and study the group $G_ au$ generated by the reflection across the edges of $ au$. In particular, we prove that the subgroup $T_ au$ of all translations in $G_ au$ is free abelian of infinite rank, while the index 2 subgroup $H_ au$ of all orientation preserving transformations in $G_ au$ is free metabelian of rank 2, with $T_ au$ as the commutator subgroup. As a consequence, the group $G_ au$ cannot be finitely presented and we provide explicit minimal infinite presentations of both $H_ au$ and $G_ au$. This answers in the affirmative the problem of the existence of a minimal presentation for the free metabelian group of rank 2. Moreover, we discuss some examples of non-trivial relations in $T_ au$ holding for given non-generic triangles $ au$.