An application of elementary real analysis to a metabelian group admitting integral polynomial exponents
An application of elementary real analysis to a metabelian group admitting integral polynomial...
Gaglione, Anthony M.; Lipschutz, Seymour; Spellman, Dennis
2015-05-01 00:00:00
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 .
http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.pngGroups Complexity Cryptologyde Gruyterhttp://www.deepdyve.com/lp/de-gruyter/an-application-of-elementary-real-analysis-to-a-metabelian-group-0vPG1JbM7p
An application of elementary real analysis to a metabelian group admitting integral polynomial exponents
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 .
To get new article updates from a journal on your personalized homepage, please log in first, or sign up for a DeepDyve account if you don’t already have one.
Access the full text.
Sign up today, get an introductory month for just $19.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.