# Thin groups and the arithmetic of imaginary quadratic fields

# Thin groups and the arithmetic of imaginary quadratic fields

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The Farey subdivision of the real line describes the action of PSL(2,Z) and gives a continued fraction algorithm approximating real numbers by rational numbers. Asmus Schmidt defined an analogue for the complex plane, depending on a choice of Euclidean imaginary quadratic field. I generalize his construction to arbitrary imaginary quadratic fields, which gives rise to an arrangement of circles into orbits of the Bianchi group PSL(2,O_K). Geometric aspects of this picture relate to arithmetic properties of the field. Furthermore, Apollonian circle packings arise naturally from the Gaussian case. The curvatures of integral Apollonian circle packings conjecturally satisfy a local-global property for which a density-one statement was proven by Bourgain and Kontorovich. Inspired by the viewpoint of Schmidt arrangements, we can define an infinite family of integral circle packings for which the methods of Bourgain-Kontorovich generalize (joint work with Elena Fuchs and Xin Zhang). Daniel Martin has recently generalized these pictures to provide a continued fraction algorithm for non-Euclidean imaginary quadratic fields.