Notes on Thin Matrix Groups

Peter Sarnak

Notes on Thin Matrix Groups

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arxiv 1212.3525 v1 pith:7PFLA3YT submitted 2012-12-14 math.NT math.GR

Peter Sarnak This is my paper

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keywords groupsmatrixnotestheorythinaffineapplicationsapproximation

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These notes were prepared for the MSRI hot topics workshop on superstrong approximation (2012). We give a brief overview of the developments in the theory, especially the fundamental expansion theorem. Applications to diophantine problems on orbits of integer matrix groups, the affine sieve, group theory, gonality of curves and Heegaard genus of hyperbolic three manifolds, are given. We also discuss the ubiquity of thin matrix groups in various contexts, and in particular that of monodromy groups.

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