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Before the talks, we will meet at 12:30 for a lunch in the cafe (MCS 0014) near the lecture theater. You will see the cafe immediately after entering the building through the main door. The cafe serves a soup and a selection of sandwiches.
In harmonic analysis spherical functions are radial functions being eigenvectors of Laplacian associated with eigenvalues. In the context of general hyperbolic groups, analytical objects such as the Laplacian are not available to consider such functions. Nevertheless we may define spherical functions for hyperbolic groups thanks to Patterson-Sullivan measures and produce unitary and non unitary representations (only Hilbertian representations in some cases). The natural question is to prove that they are irreducible though. We will discuss spectral inequalities, ergodic theorems "à la von Neumann" and analog of Riesz operators (or Knapp-Stein operators) for general hyperbolic groups.
Consider a random walk on a finitely generated group which satisfies a so-called ratio limit theorem. The limit function gives rise to a kernel, the "ratio limit kernel". That kernel is ρ-harmonic, where ρ is the spectral radius of the random walk.
Stimulated by a question posed by Adam Dor-On, the question is addressed whether the compactification of the group induced by that kernel coincides with the ρ-Martin compactification.
The anser is "yes" when the group is Gromov-hyperbolic and the random walk is symmetric with finite range. The boundary in the compactification is then the hyperbolic boundary.
[appeared in Documenta Math. 26 (2021), 1501-1528]
15:45-16:15 Tea
I will discuss the growth of the number of infinite dihedral subgroups of lattices in PSL(2, R). Such subgroups exist whenever the lattice has 2-torsion and they are closely related to so-called reciprocal geodesics on the corresponding quotient orbifold. These are closed geodesics passing through the order two orbifold point, or equivalently, homotopy classes of closed curves having a representative in the fundamental group thatâ€™s conjugate to its own inverse. We obtain the asymptotic growth of infinite dihedral subgroups (or reciprocal geodesics) in any lattice, generalizing earlier work of Sarnak and Bourgain-Kontorivich on the growth of the number of reciprocal geodesics on the modular surface. Time allowing, I will explain how our methods also show that reciprocal geodesics are equidistributed in the unit tangent bundle. This is joint work with Juan Souto.
There will be a dinner in the Akarsu Turkish Restraurant. Please e-mail Anna Felikson if you plan to attend the dinner.
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The lecture theater MCS0001 is on the ground floor of the new Maths and Computer Science building, to the left from the main enterance.