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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... 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.png Groups Complexity Cryptology de Gruyter

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

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Publisher
de Gruyter
Copyright
Copyright © 2015 by the
ISSN
1867-1144
eISSN
1869-6104
DOI
10.1515/gcc-2015-0004
Publisher site
See Article on Publisher Site

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 .

Journal

Groups Complexity Cryptologyde Gruyter

Published: May 1, 2015

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