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Uspekhi Mat. Nauk, 2019, Volume 74, Issue 3(447), Pages 17–62 (Mi umn9886)  

Conway topograph, $\mathrm{PGL}_2(\pmb{\mathbb Z})$-dynamics and two-valued groups

V. M. Buchstabera, A. P. Veselovbc

a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
b Lomonosov Moscow State University
c Loughborough University, Loughborough, UK

Abstract: Conway's topographic approach to binary quadratic forms and Markov triples is reviewed from the point of view of the theory of two-valued groups. This leads naturally to a new class of commutative two-valued groups, which we call involutive. It is shown that the two-valued group of Conway's lax vectors plays a special role in this class. The group $\mathrm{PGL}_2(\mathbb Z)$ describing the symmetries of the Conway topograph acts by automorphisms of this two-valued group. Binary quadratic forms are interpreted as primitive elements of the Hopf 2-algebra of functions on the Conway group. This fact is used to construct an explicit embedding of the Conway two-valued group into $\mathbb R$ and thus to introduce a total group ordering on it. The two-valued algebraic involutive groups with symmetric multiplication law are classified, and it is shown that they are all obtained by the coset construction from the addition law on elliptic curves. In particular, this explains the special role of Mordell's modification of the Markov equation and reveals its connection with two-valued groups in $K$-theory. The survey concludes with a discussion of the role of two-valued groups and the group $\mathrm{PGL}_2(\mathbb Z)$ in the context of integrability in multivalued dynamics.
Bibliography: 104 titles.

Keywords: Conway topograph, modular group, two-valued groups, algebraic discrete-time dynamics, integrability.

Funding Agency Grant Number
Ministry of Science and Higher Education of the Russian Federation 1.13560.2019/13.1
The research of the first author was carried out within the framework of a state assignment of the Ministry of Science and Higher Education of the Russian Federation (project no. 1.13560.2019/13.1).


DOI: https://doi.org/10.4213/rm9886

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English version:
Russian Mathematical Surveys, 2019, 74:3, 387–430

Bibliographic databases:

UDC: 511.5+512.5+517.9
MSC: Primary 11H55, 20N20; Secondary 37P99
Received: 31.01.2019

Citation: V. M. Buchstaber, A. P. Veselov, “Conway topograph, $\mathrm{PGL}_2(\pmb{\mathbb Z})$-dynamics and two-valued groups”, Uspekhi Mat. Nauk, 74:3(447) (2019), 17–62; Russian Math. Surveys, 74:3 (2019), 387–430

Citation in format AMSBIB
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\paper Conway topograph, $\mathrm{PGL}_2(\pmb{\mathbb Z})$-dynamics and two-valued groups
\jour Uspekhi Mat. Nauk
\yr 2019
\vol 74
\issue 3(447)
\pages 17--62
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\jour Russian Math. Surveys
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\vol 74
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\pages 387--430
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