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SUMMARY:Character varieties of random groups - Oren Becker\, Cambridge
DTSTART:20220209T160000Z
DTEND:20220209T170000Z
UID:TALK167140@talks.cam.ac.uk
CONTACT:Henry Wilton
DESCRIPTION:The space Hom(Γ\,G) of homomorphisms from a finitely-generate
 d group Γ to a complex semisimple algebraic group G is known as the G-rep
 resentation variety of Γ. We study this space when G is fixed and Γ is a
  random group in the few-relators model. That is\, Γ is generated by k el
 ements 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(Γ\,G)\, consisting of all homomorphisms whose images are Zariski den
 se in G. We give an explicit formula for the dimension of Z\, valid with p
 robability tending to 1\, and study the Galois action on its geometric com
 ponents. In particular\, we show that in the case of deficiency 1 (i.e.\, 
 k-r=1)\, the Zariski-dense G-representations of a typical Γ enjoy Galois 
 rigidity.\n\nOur methods assume the Generalized Riemann Hypothesis and exp
 loit mixing of random walks and spectral gap estimates on finite groups.\n
 \nBased on a joint work with E. Breuillard and P. Varju.
LOCATION:MR13
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