## Bulletin of the Belgian Mathematical Society - Simon Stevin

- Bull. Belg. Math. Soc. Simon Stevin
- Volume 25, Number 5 (2018), 653-676.

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

Stefano Isola and Riccardo Piergallini

#### 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 $\tau$.

#### Article information

**Source**

Bull. Belg. Math. Soc. Simon Stevin, Volume 25, Number 5 (2018), 653-676.

**Dates**

First available in Project Euclid: 18 January 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1547780427

**Digital Object Identifier**

doi:10.36045/bbms/1547780427

**Mathematical Reviews number (MathSciNet)**

MR3901838

**Zentralblatt MATH identifier**

07038544

**Subjects**

Primary: 20F05: Generators, relations, and presentations 20F55: Reflection and Coxeter groups [See also 22E40, 51F15] 51F15: Reflection groups, reflection geometries [See also 20H10, 20H15; for Coxeter groups, see 20F55]

**Keywords**

generic triangle typical triangle stable sequence reflection group triangle group

#### Citation

Isola, Stefano; Piergallini, Riccardo. On the generic triangle group and the free metabelian group of rank 2. Bull. Belg. Math. Soc. Simon Stevin 25 (2018), no. 5, 653--676. doi:10.36045/bbms/1547780427. https://projecteuclid.org/euclid.bbms/1547780427