On the generic triangle group

TL;DR

This paper studies the structure of reflection groups generated by a generic Euclidean triangle, revealing their complex algebraic properties and providing explicit infinite presentations, thus solving a longstanding problem in group theory.

Contribution

It introduces the concept of a generic Euclidean triangle and characterizes the algebraic structure of the associated reflection groups, including explicit minimal infinite presentations.

Findings

The translation subgroup is free abelian of infinite rank.

The orientation-preserving subgroup is free metabelian of rank 2.

The group cannot be finitely presented.

Abstract

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