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SUMMARY:Barycenters and coycles on the Furstenberg boundary - Michelle Buc
 her (Université de Genève)
DTSTART:20250709T091500Z
DTEND:20250709T101500Z
UID:TALK232837@talks.cam.ac.uk
DESCRIPTION:Let G be a semisimple\, connected\, finite center Lie group. O
 ur main result is that every continuous cohomology class on G can be repre
 sented by a continuous cocycle on an explicit open dense subset of product
 s of the Furstenberg boundary. As an application\, we will see the validit
 y of a conjecture of Monod on the injectivity of the comparison map betwee
 n bounded and unbounded cohomology in the particular case of degree 4 for 
 the connected component of the isometry group of hyperbolic n-space\, whic
 h was previously only known for n=2.&nbsp\;\nOne of the main tool is the e
 xistence of a continuous G-equivariant barycenter map from generic triples
  of points in the Furstenberg boundary into the symmetric space. I will de
 scribe our construction\, which is explicit and purely algebraic\, in the 
 simpler case when the action of the longest element of the Weyl group on t
 he Lie algebra of a maximal torus A is by -1. In the case of real hyperbol
 ic n-space we recover the geometric barycenter of the corresponding ideal 
 triangle\, but in higher rank the geometric interpration of our barycenter
  remains mysterious.&nbsp\;\nThis is joint work with Alessio Savini.&nbsp\
 ;
LOCATION:Seminar Room 1\, Newton Institute
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