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SUMMARY:Alex Lubotzky (Hebrew University of Jerusalem)
DTSTART:20210114T150000Z
DTEND:20210114T160000Z
DTSTAMP:20260423T021324Z
UID:GOThIC/12
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/GOThIC/12/">
 Stability\, non-approximated groups and high-dimensional expanders</a>\nby
  Alex Lubotzky (Hebrew University of Jerusalem) as part of GOThIC - Ischia
  Online Group Theory Conference\n\n\nAbstract\nSeveral well-known open que
 stions\, such as: "are all groups sofic or hyperlinear?"\, have a common f
 orm: can all groups be approximated \nby asymptotic homomorphisms into the
  symmetric groups $\\mathrm{Sym}(n)$ (in the sofic case) or the  unitary g
 roups $U(n)$ (in the hyperlinear case)?\n\n      In the case of $U(n)$\, t
 he question can be asked with respect to different metrics and norms. \n  
     We answer\, for the first time\, some of these versions\, showing that
  there exist finitely presented groups which are  not approximated by $U(n
 )$ with respect to the Frobenius ($=L_2$) norm and many other norms.\n\n  
   The strategy is via the notion of "stability": Some higher dimensional c
 ohomology vanishing phenomena is proven to imply stability. Using Garland 
 method  ( a.k.a. high dimensional expanders as quotients of Bruhat-Tits bu
 ildings)\, it is shown that  some non-residually-finite groups   are stabl
 e and hence cannot be approximated. These groups are  central extensions o
 f some lattices in p-adic Lie groups (constructed via  a p-adic version of
  a result of Deligne).\n\n      All notions will be explained.       Based
  on joint works with M. De Chiffre\, L. Glebsky and A. Thom and with I. Op
 penheim .\n
LOCATION:https://researchseminars.org/talk/GOThIC/12/
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