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- Title
On the generic triangle group and the free metabelian group of rank 2.
- Authors
Isola, Stefano; Piergallini, Riccardo
- Abstract
We introduce the concept of a generic Euclidean triangle τ and study the group Gτ generated by the reflection across the edges of τ. In particular, we prove that the subgroup Tτ of all translations in Gτ is free abelian of infinite rank,while the index 2 subgroup Hτ of all orientation preserving transformations in Gτ is free metabelian of rank 2, with Tτ as the commutator subgroup. As a consequence, the group Gτ cannot be finitely presented and we provide explicit minimal infinite presentations of both Hτ and Gτ. 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 Tτ holding for given non-generic triangles τ.
- Subjects
FREE metabelian groups; EUCLIDEAN geometry; TRIANGLES; ANGLES; BAIRE classes
- Publication
Bulletin of the Belgian Mathematical Society - Simon Stevin, 2018, Vol 25, Issue 5, p653
- ISSN
1370-1444
- Publication type
Article
- DOI
10.36045/bbms/1547780427