Geodesic laminations and noncommutative geometry

Measured geodesic laminations is a remarkable abstraction (due to W. P. Thurston) of many otherwise unrelated phenomena occurring in differential geometry, complex analysis and geometric topology. In this article we focus on connections of geodesic laminations with the inductive limits of finite-dimensional semi-simple C∗-algebras (AF C∗-algebras). Our main result is a bijection between combinatorial presentation of such C∗-algebras (so-called Bratteli diagrams) and measured geodesic laminations on compact surfaces. This link appears helpful indeed as it provides insights to the Teichmüller spaces, Thurston’s theory of surface homeomorphisms, Stallings’ fibrations to the one side, and noncommutative (algebraic) geometry to the other.