On the generic triangle group

2014/05/08 by Stefano Isola, Isola, Stefano, Riccardo Piergallini +1
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR) #Metric Geometry (math.MG) #math.DS #math.GR #math.MG

paper · pdf · doi:10.48550/arxiv.1405.1881

21 pages, 6 figures

arxiv created 2015/06/25 · arxiv updated 2015/06/26

Abstract

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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