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SUMMARY:Random covering spaces of closed surfaces are near-optimal expande
 rs - Michael Magee (Durham)
DTSTART:20260211T160000Z
DTEND:20260211T170000Z
UID:TALK242890@talks.cam.ac.uk
CONTACT:Oscar Randal-Williams
DESCRIPTION:Random covering spaces of a wedge sum of circles give us rando
 m regular graphs\, a much-studied class of objects. For instance\, a theor
 em of Friedman tells us that with high probability they are near-optimal e
 xpander graphs. \n\nI'm going to talk about the extension of this fact to 
 covering spaces of closed surfaces. The notion of `expander' surface can b
 e either combinatorial or Riemannian.  In the Riemannian case we rely on a
   vast  strengthening of near-optimal expander graphs known as `strong con
 vergence'.\n\nThe launching off point is a study of the statistics of lift
 ing curves from closed surfaces to uniformly randomly chosen degree-n cove
 ring spaces\, and the fact that certain functions appearing here are `almo
 st rational' functions of the degree of the covering spaces.\n\nThis is co
 mbined with recent technology (the `polynomial method') for establishing e
 xpansion properties of random graphs\, and an interesting (new) fact about
  random walks on surface groups.\n\nBased on joint works with Hide\, Puder
 \, de la Salle\, and van Handel.
LOCATION:MR13
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