An application of elementary real analysis to a metabelian group admitting integral polynomial exponents

Abstract

Abstract Let G be a free metabelian group of rank r = 2. We introduce a faithful 2×2 real matrix representation of G and extend this to a group G ℤ[θ] $G^{\mathbb {Z}[\theta ]}$ of 2×2 matrices admitting exponents from the integral polynomial ring ℤ[θ]$\mathbb {Z}[\theta ]$ . Identifying G with its matrix representation, we show that given γ(θ)∈G ℤ[θ] $\gamma (\theta )\in G^{\mathbb {Z}[\theta ]}$ and n∈ℤ$n\in \mathbb {Z}$ , one has that lim θ→n γ(θ)$\lim _{\theta \rightarrow n}\gamma (\theta )$ exists and lies in G. Furthermore, the maps γ(θ)↦lim θ→n γ(θ)$\gamma (\theta )\mapsto \lim _{\theta \rightarrow n}\gamma (\theta )$ form a discriminating family of group retractions G ℤ[θ] →G$G^{\mathbb {Z}[\theta ]}\rightarrow G$ as n varies over ℤ. Although not explicitly carried out in this manuscript, it is clear that similar results hold for any countable rank r.