[Paper Review] Asymptotic Cohomology and Uniform Stability for Lattices in Semisimple Groups

Lev Glebsky, Alexander Lubotzky+2 more|arXiv (Cornell University)|Jan 1, 2023

Advanced Operator Algebra Research4 citations

TL;DR

This paper establishes uniform stability for high-rank lattices in semisimple groups with respect to unitary representations on finite-dimensional Hilbert spaces under submultiplicative norms. By developing an asymptotic cohomology theory that captures stability obstructions, the authors prove that vanishing of the second asymptotic cohomology implies uniform stability, extending Kazhdan's result for amenable groups and Burger-Ozawa-Thom's result for $SL(n,\mathbb{Z})$ ($n>2$).

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: 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.

Motivation & Objective

• To establish a new rigidity property—uniform stability—for high-rank lattices in semisimple groups.
• To extend known results on Ulam stability from amenable groups and $SL(n,\mathbb{Z})$ to general higher-rank lattices.
• To develop a cohomological framework—'asymptotic cohomology'—that captures the obstruction to uniform stability.
• To clarify the relationship between bounded cohomology, asymptotic cohomology, and Ulam stability, particularly in the context of vanishing second cohomology.

Proposed method

• Introduce a new cohomology theory—'asymptotic cohomology'—as an asymptotic variant of bounded cohomology to study stability obstructions.
• Define the asymptotic cohomology group $\mathrm{H}_a^2(\Gamma, \mathcal{W})$ for a lattice $\Gamma$ and a family of Banach $\Gamma$-modules $\mathcal{W}$, modeling finite-dimensional unitary representations.
• Prove that vanishing of $\mathrm{H}_a^2(\Gamma, \mathcal{W})$ implies uniform stability of $\Gamma$ with respect to the family of unitary groups on finite-dimensional Hilbert spaces.
• Establish a comparison map from asymptotic cohomology to bounded cohomology $\mathrm{H}_b^2(\Gamma, \widetilde{\mathcal{W}})$, linking the new theory to well-studied bounded cohomological invariants.
• Use the structure of parabolic subgroups and the vanishing of bounded cohomology in higher-rank lattices to show that $\mathrm{H}_a^2(\Gamma, \mathcal{W}) = 0$ for such groups.
• Leverage Margulis super-rigidity and arithmeticity to reduce the study of representations to Galois-twisted and finite quotient representations, enabling cohomological control.

Experimental results

Research questions

• RQ1 Does the vanishing of second asymptotic cohomology imply uniform stability for lattices in higher-rank semisimple groups?
• RQ2 Can asymptotic cohomology serve as a bridge between bounded cohomology and Ulam stability in the context of finite-dimensional unitary representations?
• RQ3 Is the property $G(\mathcal{Q}_1, \mathcal{Q}_2)$ necessary for the stability result, or is it a technical artifact of the proof?
• RQ4 To what extent does the comparison map $\mathrm{H}_a^2(\Gamma, \mathcal{W}) \to \mathrm{H}_b^2(\Gamma, \widetilde{\mathcal{W}})$ capture the stability obstruction?
• RQ5 Can the framework of asymptotic cohomology be extended to non-uniform lattices or non-archimedean settings?

Key findings

• The second asymptotic cohomology group $\mathrm{H}_a^2(\Gamma, \mathcal{W})$ vanishes for all high-rank lattices $\Gamma$ in semisimple groups, implying uniform stability.
• Uniform stability holds for all high-rank lattices $\Gamma$ with respect to unitary representations on finite-dimensional Hilbert spaces under submultiplicative norms.
• The vanishing of $\mathrm{H}_a^2(\Gamma, \mathcal{W})$ is sufficient for uniform stability, generalizing Kazhdan's result for amenable groups.
• The comparison map $\mathrm{H}_a^2(\Gamma, \mathcal{W}) \to \mathrm{H}_b^2(\Gamma, \widetilde{\mathcal{W}})$ is not necessarily injective or surjective, but its kernel captures the stability obstruction.
• For lattices in higher-rank simple Lie groups, $\mathrm{H}_b^2(\Gamma, W) = 0$ for all dual separable Banach $\Gamma$-modules $W$, suggesting a potential link to asymptotic cohomology.
• The framework distinguishes uniform from pointwise stability, showing that uniform stability holds even when pointwise stability fails under the operator norm, particularly for $p = \infty$.