Matrix theoretic interpretation of the classical Markoff theory (Q1854143)

This work explores H. Cohn's approach to the integer solutions of Markoff's equation: \(m^2 + m_{1}^2 + m_{2}^2 = 3 m m_1 m_2 \). Cohn showed that the solutions correspond to appropriately normalized traces of triplets \((A, B, AB)\) in \(\text{SL}(2, \mathbb Z)\) such that \(A\) and \(B\) generate the commutator group of \(\text{SL}(2, \mathbb Z)\), a free group on two generators. After a particularly clear introduction to Cohn's approach, the author gives a detailed analysis of related explicit representations of automorphisms of the free group on two generators.