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CATEGORIES:Differential Geometry and Topology Seminar
SUMMARY:Character varieties of random groups - Oren Becker
\, Cambridge
DTSTART;TZID=Europe/London:20220209T160000
DTEND;TZID=Europe/London:20220209T170000
UID:TALK167140AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/167140
DESCRIPTION:The space Hom(Γ\,G) of homomorphisms from a finite
ly-generated group Γ to a complex semisimple algeb
raic group G is known as the G-representation vari
ety of Γ. We study this space when G is fixed and
Γ is a random group in the few-relators model. Tha
t is\, Γ is generated by k elements subject to r r
andom relations of length L\, where k and r are fi
xed and L tends to infinity.\n\nMore precisely\, w
e 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 dimen
sion of Z\, valid with probability tending to 1\,
and study the Galois action on its geometric compo
nents. 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 rigid
ity.\n\nOur methods assume the Generalized Riemann
Hypothesis and exploit 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
CONTACT:Henry Wilton
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