TY - JOUR
T1 - Asymptotic Cohomology and Uniform Stability for Lattices in Semisimple Groups
AU - Glebsky, Lev
AU - Lubotzky, Alexander
AU - Monod, Nicolas
AU - Rangarajan, Bharatram
N1 - The second author acknowledges with gratitude the hospitality and support of the Fields Institute (Toronto) and the Institute for Advanced Study (Princeton) where part of this work was carried on, as well as a grant by the European Research Council (ERC) under the European Union’s Horizon 2020 (grant agreement No 882751). The results presented here are part of the fourth author’s PhD thesis, also supported by the same grant. The authors would like to thank Andrei Rapinchuk for his guidance in the proof of Proposition 6.1.1. This paper is dedicated to the memory of Bob Zimmer in honor of his remarkable achievements and influence on the study of rigidity of lattices in semisimple Lie groups.
PY - 2023/1/1
Y1 - 2023/1/1
N2 - 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: uniform stability with respect to the family of unitary operators on finite-dimensional Hilbert spaces equipped with submultiplicative norms. Namely, we show that for (most) high-rank lattices, every finite-dimensional unitary "almost-representation" of Γ is a small deformation of a (true) unitary representation. This extends a result of Kazhdan (1983) for amenable groups and of Burger-Ozawa-Thom (2013) for SL(n,Z) (for n>2). 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 question raised in Monod (2006) about a possible connection between vanishing of second bounded cohomology and Ulam stability.
AB - 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: uniform stability with respect to the family of unitary operators on finite-dimensional Hilbert spaces equipped with submultiplicative norms. Namely, we show that for (most) high-rank lattices, every finite-dimensional unitary "almost-representation" of Γ is a small deformation of a (true) unitary representation. This extends a result of Kazhdan (1983) for amenable groups and of Burger-Ozawa-Thom (2013) for SL(n,Z) (for n>2). 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 question raised in Monod (2006) about a possible connection between vanishing of second bounded cohomology and Ulam stability.
U2 - 10.48550/arXiv.2301.00476
DO - 10.48550/arXiv.2301.00476
M3 - مقالة
SN - 2331-8422
JO - arxiv.org
JF - arxiv.org
ER -