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SUMMARY:Character varieties of random groups - Oren Becker (University of 
 Cambridge)
DTSTART:20211119T134500Z
DTEND:20211119T144500Z
UID:TALK164320@talks.cam.ac.uk
CONTACT:76015
DESCRIPTION:The space Hom(\\Gamma\,G) of homomorphisms from a finitely-gen
 erated group \\Gamma to a complex semisimple algebraic group G is known as
  the G-representation variety of \\Gamma. We study this space when G is fi
 xed and \\Gamma is a random group in the few relators model. That is\, \\G
 amma is generated by k elements subject to r random relations of length L\
 , where k and r are fixed and L tends to infinity.\n\nMore precisely\, we 
 study the subvariety Z of Hom(\\Gamma\,G)\, consisting of all homomorphism
 s whose images are Zariski dense in G. We give an explicit formula for the
  dimension of Z\, valid with probability tending to 1\, and study the Galo
 is action on its geometric components. In particular\, we show that in the
  case of deficiency 1 (i.e.\, k-r=1)\, the Zariski-dense G-representations
  of a typical \\Gamma enjoy Galois rigidity.\n\nOur methods assume the Gen
 eralized Riemann Hypothesis and exploit mixing of random walks and spectra
 l gap estimates on finite groups.\n\nBased on a joint work with E. Breuill
 ard and P. Varju.\n
LOCATION:In person if possible
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