On the generic triangle group

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

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