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Free subgroups of linear groups and diophantine heights
If you have a question about this talk, please contact Ben Green.
I will first discuss the statement and some ideas of the proof of Tits’ alternative for groups generated by finitely many matrices over an arbitrary field. Then I will present a strengthening of Tits’ theorem which can be described as a uniform effective version of it. This strong Tits alternative has various consequences regarding the growth, co-growth and uniform spectral gap of finitely generated linear groups. Reducing mod p it also yield some new results about the girth of finite subgroups of GL_d(Z/pZ). Underlying the proofs are some number theoretical results about points of small height in algebraic varieties and an unexpected connection with the Lehmer Conjecture
This talk is part of the Discrete Analysis Seminar series.
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