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SUMMARY:Nilpotent approximate groups - Matthew Tointon (Cambridge)
DTSTART:20121031T160000Z
DTEND:20121031T170000Z
UID:TALK41200@talks.cam.ac.uk
CONTACT:Ben Green
DESCRIPTION:A fundamental theorem of additive combinatorics is Freiman's t
 heorem\,\nproved in the 1960s\, the statement of which is roughly as follo
 ws.\nSuppose a set A of integers has the property that the number of integ
 ers\nthat can be expressed as the sum of two (not necessarily distinct)\nm
 embers of A is 'not much greater' than the cardinality of A. Then\nFreiman
 's theorem says that A is contained inside a low-dimensional\ngeneralised 
 arithmetic progression of cardinality not much greater than\nthat of A. In
  recent years Green and Ruzsa generalised this result to\nall abelian grou
 ps\, where the conclusion is that A is efficiently\ncontained inside the s
 um of a finite subgroup and a low-dimensional\nprogression.\n\nIn non-abel
 ian groups the analogue of the hypothesis of Freiman's\ntheorem is that A 
 is an 'approximate group'\, which roughly means that\nthe set of all eleme
 nts of the form xy\, with x and y belonging to A\, can\nbe covered by a fe
 w translates of A. Generalising Freiman's theorem in a\ndifferent directio
 n\, Breuillard and Green have shown that approximate\nsubgroups inside any
  torsion-free nilpotent group can be controlled by\nprogressions\, provide
 d that the notion of progression is suitably\nmodified from the abelian ve
 rsion.\n\nIn this talk I will give an outline of a result that generalises
  all of\nthese statements. Specifically\, this result says that an approxi
 mate\nsubgroup A of an arbitrary nilpotent group G is efficiently containe
 d in\nthe product of a finite subgroup normalised by A and a nilpotent\npr
 ogression. I will probably specialise to the case in which G is of\nnilpot
 ency class 2\, in which the ideas of the argument are all present\nbut the
  technical details are far simpler than in general. If there is\ntime at t
 he end I will give some pointers on how to proceed to the\ngeneral case.\n
LOCATION:MR11\, CMS
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