## Abstract

Conways 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 Conways lax vectors plays a special role in this class. The group PGL2(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 R and thus to introduce a total group ordering on it. The two-valued algebraic involutive groups with symmetric multiplication law are classifed, 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 Mordells modifcation 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 PGL2(Z) in the context of integrability in multivalued dynamics.

Original language | English |
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Pages (from-to) | 387-430 |

Number of pages | 44 |

Journal | Russian Mathematical Surveys |

Volume | 74 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2019 |

Externally published | Yes |

## Keywords

- algebraic discrete-time dynamics
- Conway topograph
- integrability
- modular group
- two-valued groups

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