Spectral gap for the cohomological Laplacian of $\operatorname{SL}_3(\mathbb{Z})$

TL;DR

This paper proves that the degree 1 cohomological Laplacian of SL_3(Z) can be expressed as a sum of hermitian squares, providing spectral gap estimates for all unitary representations, advancing understanding of its algebraic and spectral properties.

Contribution

It demonstrates that the cohomological Laplacian in degree 1 for SL_3(Z) is a sum of hermitian squares and offers spectral gap estimates across all unitary representations.

Findings

Laplacian expressed as sum of hermitian squares

Spectral gap estimates provided for all unitary representations

Enhanced understanding of algebraic structure of SL_3(Z) cohomology

Abstract

We show that the cohomological Laplacian in degree 1 in the group cohomology of $\operatorname{SL}_3(\mathbb{Z})$ 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.