Conway topograph, PGL(2)(Z)-dynamics and two-valued groups

Conway’s topographic approach to the binary quadratic forms and Markov triples is reviewed from the point of view of the theory of two-valued groups. This naturally leads to a new class of commutative two-valued groups, which we call involutive. We show that the two-valued group of Conway’s lax vectors plays a special role in this class. The group P GL2(Z), describing the symmetries of Conway topograph, acts by the automorphisms of this two-valued group. The binary quadratic forms are interpreted as primitive elements of 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 the total group ordering on it. We classify all two-valued algebraic involutive groups with symmetric multiplication law and show that they are given by the coset construction from the addition law on the elliptic curves. In particular, this explains a special role of Mordell’s modification of the Markov equation and reveals its relation to the two-valued group from K-theory. We finish with the discussion of the role of two-valued groups and P GL2(Z) in the context of integrability in multi-valued dynamics.