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Groups Complexity Cryptology
, Volume 7 (1) – May 1, 2015

/lp/de-gruyter/an-application-of-elementary-real-analysis-to-a-metabelian-group-0vPG1JbM7p

- 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 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 .

Groups Complexity Cryptology – de Gruyter

**Published: ** May 1, 2015

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