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Universität Augsburg
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Universität Augsburg
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Professor Dr. Chiranjib Mukherjee
WWU Münster
spricht am
Mittwoch, 12. Juli 2023
um
16:00 Uhr
im
Raum 1010 (L1)
über das Thema:
| Abstract: |
| Let G be the Cayley graph of a finitely generated group Γ. We show that Γ has the Haagerup property if and only if there is a Γ-invariant bond (or site) percolation with its marginals approaching one and with its the two-point function vanishing at infinity. Our result is inspired by the characterization of amenability by Benjamini, Lyons, Peres and Schramm. Our proof is based on a new construction using invariant point processes on spaces with measured walls, leading to quantitative bounds on the two-point functions (e.g. it yields exponential decay of the two-point function in several examples, including co-compact Fuchsian groups and lamplighters over free groups). Moreover, our method allows us to strengthen a consequence of Kazhdan's property (T), due to Lyons and Schramm, to an equivalence. Namely, we show that Γ has property (T) if and only if for every Γ-invariant bond percolation, large marginals imply that the two-point function is bounded away from zero. We extend this result to the setting of relative property (T) and use the corresponding threshold to give a new proof of the fact, already observed by Gaboriau and Tucker-Drob, that there is no unique infinite cluster at the uniqueness threshold for Bernoulli bond percolation on Cayley graphs of groups admitting an infinite normal subgroup with relative property (T). Joint work with Konstantin Recke (Münster). |
| Hierzu ergeht herzliche Einladung. |
| Prof. Dr. Stefan Großkinsky |