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Asymptotic Cohomology and Uniform Stability for Lattices in Semisimple Groups | | Lev Glebsky
; Alexander Lubotzky
; Nicolas Monod
; Bharatram Rangarajan
; | | Date: |
1 Jan 2023 | | Abstract: | It is, by now, classical that lattices in higher rank semisimple groups have
various rigidity properties. In this work, we add another such rigidity
property to the list, namely uniformly stability with respect to the family of
unitary operators on finite-dimensional Hilbert spaces equipped with
submultiplicative norms. Towards this goal, we first build an elaborate
cohomological theory capturing the obstruction to such stability, and show that
the vanishing of second cohomology implies uniform stability in this setting.
This cohomology can be roughly thought of as an asymptotic version of bounded
cohomology, and sheds light on a possible connection between vanishing of
second bounded cohomology and Ulam stability. Along the way, we use this
criterion to provide a short conceptual (re)proof of the classical result of
Kazhdan that discrete amenable groups are Ulam stable. We then use this
machinery to establish our main result, that lattices in a class of higher rank
semisimple groups (which are known to have vanishing bounded cohomology) are
uniformly stable. | | Source: | arXiv, 2301.00476 | | Services: | Forum | Review | PDF | Favorites | |
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