Spectral Gap for the Cohomological Laplacian of SL₃(ℤ)

Abstract

We show that the cohomological Laplacian in degree 1 in the group cohomology of ${\text{SL}}_3(\mathbb{Z})$ is a sum of hermitian squares in the algebra ${\mathbb{M}}_n(\mathbb{R}G)$. We provide an estimate of the spectral gap for this Laplacian for every unitary representation. Our results have been obtained by convex optimization technique of semidefinite programming (SDP).

KEYWORDS:

• property (T)
• semidefinite programming
• group cohomology
• Laplacian

Additional information

Funding

The first author was supported by SPP 2026 Geometry at infinity funded by the Deutsche Forschungsgemeinschaft. The second and third authors were supported by the European Research Council (ERC) grant Rigidity of groups and higher index theory under the European Union’s Horizon 2020 research and innovation program (grant agreement no. 677120-INDEX).