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# Operator Algebra and Noncommutative Geometry Seminar Series

## February 15, 2018 @ 3:30 pm - 4:30 pm

##### Speaker

Professor Bram Mesland (University of Bonn)

##### Title

Integral operators and the noncommutative boundary of hyperbolic manifolds

##### Abstract

The fundamental group of a hyperbolic manifold M can be realized as a group of Moebius transformations on the hyperbolic ball. As such it also acts on the sphere (the boundary of the ball) and it gives rise to a purely infinite crossed product C*-algebra. By a result of Connes, spectral triples on such algebras are elusive in the sense that they cannot be finitely summable. In these talks I will describe how Dirac operators on hypersurfaces in M can be pushed forward to spectral triples on the boundary crossed product algebra by means of a singular integral operator. The latter is built from the field of harmonic measures on the hyperbolic ball and is the formal analogue of the logarithm of the Laplacian on the sphere.