Arithmetic Subgroups and Applications

Arithmetic Subgroups and Applications

Mariam Almahdi Mohammed Mulla

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Amal Mohammed Ahmed Gaweash

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Hayat Yousuf Ismail Bakur

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University of Hafr Al-Batin

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Abstract

Arithmetic subgroups are an important source of discrete groups acting freely on manifolds. We need to know that there exist many torsion-free 𝑺𝑺L(𝟐𝟐, ℝ) is an “arithmetic” subgroup of 𝑺𝑺L(𝟐𝟐, ℝ). The other arithmetic subgroups are not as obvious, but they can be constructed by using quaternion algebras. Replacing the quaternion algebras with larger division algebras yields many arithmetic subgroups of 𝑺𝑺L(𝒏𝒏, ℝ), with 𝒏𝒏>2. In fact, a calculation of group cohomology shows that the only other way to construct arithmetic subgroups of 𝑺𝑺L(𝒏𝒏, ℝ) is by using arithmetic groups. In this paper justifies Commensurable groups, and some definitions and examples,ℝ-forms of classical simple groups over ℂ, calculating the complexification of each classical group, Applications to manifolds. Let us start with 𝑺𝑺𝑺𝑺(𝑛𝑛,ℂ). This is already a complex Lie group, but we can think of it as a real Lie group of twice the dimension. As such, it has a complexification.

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Funding

No external funding was declared for this work.

Conflict of Interest

The authors declare no conflict of interest.

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How to Cite This Article

Mariam Almahdi Mohammed Mulla. 2020. \u201cArithmetic Subgroups and Applications\u201d. Global Journal of Science Frontier Research - F: Mathematics & Decision GJSFR-F Volume 20 (GJSFR Volume 20 Issue F6).

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GJSFR Volume 20 Issue F6

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Crossref Journal DOI 10.17406/GJSFR

Print ISSN 0975-5896

e-ISSN 2249-4626

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GJSFR-F Classification MSC 2010: 03C62

Submission ReceivedJuly 30, 2020
Peer Review Double Blind
Handling Editor
Accepted August 19, 2020
Published September 2, 2020

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September 30, 2020

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