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The Modular Group SL ( $$2, \mathbb{Z}$$ )

In: Markov's Theorem and 100 Years of the Uniqueness Conjecture

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  • Martin Aigner

    (Freie Universität Berlin, Fachbereich Mathematik und Informatik Institut für Mathematik)

Abstract

The last two chapters contained the basic combinatorial datasets for the study of Markov numbers: Markov tree, Farey tree, and Cohn tree. In this chapter and the next, we take an algebraic point of view. The Cohn matrices are elements of the modular group, and this group plays a central role in several different and very interesting settings. So, we will suspect that studying these connections will also shed new light on the Markov numbers and the uniqueness conjecture. This is indeed the case: The next two chapters contain some of the most beautiful results on the way to Markov’s theorem. We study the full modular group SL( $$2, \mathbb{Z}$$ ) in this chapter and the subgroup generated by the Cohn matrices in the next.

Suggested Citation

  • Martin Aigner, 2013. "The Modular Group SL ( $$2, \mathbb{Z}$$ )," Springer Books, in: Markov's Theorem and 100 Years of the Uniqueness Conjecture, edition 127, chapter 5, pages 81-111, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-00888-2_5
    DOI: 10.1007/978-3-319-00888-2_5
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