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SUMMARY:Dense and sparse unique infinite clusters in Kazhdan groups - Kons
 tantin Recke (University of Oxford)
DTSTART:20251009T091500Z
DTEND:20251009T101500Z
UID:TALK236257@talks.cam.ac.uk
DESCRIPTION:Bond percolations on Cayley graphs are probabilistic processes
  in which edges are deleted randomly in an invariant way. In this talk\, w
 e will consider the question about the existence of a unique infinite clus
 ter\, which has deep links to geometry\, group theory and ergodic theory. 
 We first show a probabilistic characterization of Kazhdan's property (T) w
 hich roughly speaking asserts that dense bond percolations form a unique i
 nfinite cluster. A complimentary result that Kazhdan groups admit some spa
 rse bond percolation with a unique infinite cluster was proved in a recent
  breakthrough of Hutchcroft and Pete as the central ingredient in establis
 hing that these groups have cost $1$. Motivated by the fixed price conject
 ure\, the question whether such models can be built as a factor of i.i.d. 
 was posed implicitly there and explicitly by Pete and Rokob. We show that 
 the answer is affirmative for co-compact lattices in connected higher rank
  semisimple real Lie groups with property (T). Our proof uses a new phenom
 enon in continuum percolation on the associated symmetric space\, which bu
 ilds on recent breakthrough results about Poisson--Voronoi tessellations b
 y Fraczyk\, Mellick and Wilkens.\n&nbsp\;\nBased on joint works with Chira
 njib Mukherjee (Munster) and with Jan Grebik (Leipzig).
LOCATION:Seminar Room 1\, Newton Institute
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