We introduce the concept of a generic Euclidean triangle Ο and study the
group GΟβ generated by the reflection across the edges of Ο. In
particular, we prove that the subgroup TΟβ of all translations in GΟβ
is free abelian of infinite rank, while the index 2 subgroup HΟβ of all
orientation preserving transformations in GΟβ is free metabelian of rank
2, with TΟβ as the commutator subgroup. As a consequence, the group
GΟβ cannot be finitely presented and we provide explicit minimal infinite
presentations of both HΟβ and GΟβ. 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
TΟβ holding for given non-generic triangles Ο.Comment: 21 pages, 6 figure