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SUMMARY:Free subgroups of linear groups and diophantine heights - Emmanuel
  Breuillard\, Ecole Polytechnique
DTSTART:20080429T150000Z
DTEND:20080429T160000Z
UID:TALK10907@talks.cam.ac.uk
CONTACT:Ben Green
DESCRIPTION:I will first discuss the statement and some ideas of the proof
 \nof Tits' alternative for groups generated by finitely many matrices over
 \nan arbitrary field. Then I will present a strengthening of Tits' theorem
 \nwhich can be described as a uniform effective version of it. This strong
 \nTits alternative has various consequences regarding the growth\, co-grow
 th\nand uniform spectral gap of finitely generated linear groups. Reducing
  mod\np it also yield some new results about the girth of finite subgroups
  of\nGL_d(Z/pZ). Underlying the proofs are some number theoretical results
 \nabout points of small height in algebraic varieties and an unexpected co
 nnection with the Lehmer Conjecture\n
LOCATION:MR13\, CMS
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