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SUMMARY: The diameter of somewhat dense Cayley graphs on $A_n$ -  Peter Ke
 evash (Oxford)
DTSTART:20260226T143000Z
DTEND:20260226T153000Z
UID:TALK243703@talks.cam.ac.uk
CONTACT:103978
DESCRIPTION:A fundamental open problem on the geometry of Cayley graphs is
  Babai's Diameter Conjecture\, which states that the diameter of any conne
 cted Cayley graph on a nonabelian finite simple group $G$ is at most polyl
 ogarithmic in $|G|$. A natural extremal variant\, also open in general\, a
 sks for the maximum possible diameter given the density of the generating 
 set. In this talk\, we consider the alternating permutation groups $A_n$\,
  for which Helfgott and Seress showed that the diameter of any Cayley grap
 h is at most quasipolynomial in $n$. We will present an essentially optima
 l upper bound on the diameter when the density of the generating set is at
  least $2^{-O(n)}$. Our proof combines combinatorial\, analytic and algebr
 aic arguments\, with the key ingredient being a new sharp hypercontractive
  inequality in $S_n$. This is joint work with Noam Lifshitz.
LOCATION:MR12
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