The Cohn Tree

Abstract

We saw in the last chapter that the neighbor relation between Markov triples can be conveniently encoded in an infinite binary tree. All early researchers from Markov onward used this device, but went only a little beyond it. In the 1950s, however, there was a major new development regarding Markov numbers and the uniqueness problem when Harvey Cohn noticed that a wellknown identity involving traces of integral $2 \times 2$ matrices looks very much like Markov’s equation. His discovery initiated a completely new approach to the Markov theme, with amazingly simple proofs of some further uniqueness results. In this chapter, we work out the precise relationship between Markov numbers and matrices and move on to a deeper study of the algebraic structure of the group generated by these matrices in the next part.