Inducing spectral gaps for the cohomological Laplacians of SL_n(ℤ) and SAut(F_n)

2024/04/16 by Piotr Mizerka, Mizerka, Piotr · 1 citation
Mathematics · #20G10 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2404.10287

openalex publication_date 2024/04/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The technique of inducing spectral gaps for cohomological Laplacians in degree zero was used by Kaluba, Kielak and Nowak to prove property (T) for SAut(F_n) and SL_n(ℤ). In this paper, we adapt this technique to Laplacians in degree one. This allows to provide a lower bound for the cohomological Laplacian in degree one for SL_n(ℤ) for every unitary representation. In particular, one gets in that way an alternative proof of property (T) for SL_n(ℤ) whenever n≥ 3.

Cited by

• Inducing spectral gaps for the cohomological Laplacians of Sp_2n(ℤ)

Related

• Inducing spectral gaps for the cohomological Laplacians of Sp_2n(ℤ)
• Spectral gap for the cohomological Laplacian of SL₃(ℤ)
• Proof of Lew's conjecture on the spectral gaps of simplicial complexes
• Certifying numerical estimates of spectral gaps
• Spectral gap bounds for the simplicial Laplacian and an application to random complexes