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Published online by Cambridge University Press: 29 May 2025
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.
The Chinese remainder theorem for SLn(ℤ) asserts, among other things, that for q ≥ 1, the reduction πq : SLn(ℤ)→SLn(ℤ/qℤ) is onto. Far less elementary is the extension of this feature to G(ℤ) where G is a suitable matrix algebraic group defined over ℚ. The general form of this phenomenon for arithmetic groups is known as strong approximation and it is well understood [Platonov and Rapinchuk 1994].
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