Notes on Thin Matrix Groups

By Peter Sarnak. 2012

88 citations recorded by the source corpus.

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.

View Paper: https://www.semanticscholar.org/paper/70157b6deb0fa0b3a59fd05a64f09454e6f48011

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1. Peter Sarnak
2. Minju M. Lee
3. Samuel Edwards
4. H. Oh
5. Christopher Brav
6. Hugh Thomas
7. Anna Wienhard
8. E. Fuchs

Citations in the corpus

• Anosov groups: local mixing, counting and equidistribution (2020, background): “, thin matrix groups in the terminology of [52]) arising from Hitchin representations, as constructed in Kac-Vinberg [23] and Long-Reid-Thislethwaite [28] (see also [57, Theorem 24]).”
• Thin monodromy in Sp(4) (2012, background): “k 1 or is a product of groups of rank 1, so it is natural to ask if there are examples of thin monodromy in higher rank. For more discussion of this problem, see Sarnak’s Notes on thin matrix groups ([12]). The simplest test cases are families over P1 nf0;1;1gwith monodromy having Zariski closure Sp”
• AN INVITATION TO HIGHER TEICHMÜLLER THEORY (2019, unknown): “Peter Sarnak (2014).”
• An invitation to higher Teichm\"uller theory (2018, background): “Such groups have been coined thin groups [101].”
• Hyperbolic monodromy groups for the hypergeometric equation and Cartan involutions (2013, background): “important in various associated number theoretic problems (see [S1]) and this paper is concerned”