Barycenters and coycles on the Furstenberg boundary

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• Michelle Bucher (Université de Genève)
• Wednesday 09 July 2025, 10:15-11:15
• Seminar Room 1, Newton Institute.

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NPCW06 - Non-positive curvature and applications

Let G be a semisimple, connected, finite center Lie group. Our main result is that every continuous cohomology class on G can be represented by a continuous cocycle on an explicit open dense subset of products of the Furstenberg boundary. As an application, we will see the validity of a conjecture of Monod on the injectivity of the comparison map between bounded and unbounded cohomology in the particular case of degree 4 for the connected component of the isometry group of hyperbolic n-space, which was previously only known for n=2.  One of the main tool is the existence of a continuous G-equivariant barycenter map from generic triples of points in the Furstenberg boundary into the symmetric space. I will describe our construction, which is explicit and purely algebraic, in the simpler case when the action of the longest element of the Weyl group on the Lie algebra of a maximal torus A is by -1. In the case of real hyperbolic n-space we recover the geometric barycenter of the corresponding ideal triangle, but in higher rank the geometric interpration of our barycenter remains mysterious.  This is joint work with Alessio Savini. 

This talk is part of the Isaac Newton Institute Seminar Series series.

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