Noncommutative Markov Numbers

Markov numbers are a family of positive integers originally studied in relation bounds on approximating irrational numbers with rationals. They are characterized as integer solutions to the equation $x^2+y^2+z^2=3xyz$. We will review the relationship between this equation and the cluster algebra structure of arcs on a punctured torus. Using this relationship we use the work of Berenstein and Retakh to associate a noncommutative cluster structure with values in any ring with involution. We will give examples of many such rings and the new Markov-like numbers found in them.