december 2018 On the generic triangle group and the free metabelian group of rank 2
Stefano Isola, Riccardo Piergallini
Bull. Belg. Math. Soc. Simon Stevin 25(5): 653-676 (december 2018). DOI: 10.36045/bbms/1547780427


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 $\tau$.


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Stefano Isola. Riccardo Piergallini. "On the generic triangle group and the free metabelian group of rank 2." Bull. Belg. Math. Soc. Simon Stevin 25 (5) 653 - 676, december 2018.


Published: december 2018
First available in Project Euclid: 18 January 2019

zbMATH: 07038544
MathSciNet: MR3901838
Digital Object Identifier: 10.36045/bbms/1547780427

Primary: 20F05 , 20F55 , 51F15

Keywords: generic triangle , Reflection group , stable sequence , triangle group , typical triangle

Rights: Copyright © 2018 The Belgian Mathematical Society


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Vol.25 • No. 5 • december 2018
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