Computer proofs for Property (T), and SDP duality

2020/09/10 by Martin Nitsche, Nitsche, Martin · 2 citations
Computer Science · Mathematics · #Advanced Optimization Algorithms Research #Complexity and Algorithms in Graphs #FOS: Mathematics #Group Theory (math.GR) #Machine Learning and Algorithms #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.2009.05134

openalex publication_date 2020/09/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that the semidefinite programs involved in the computer proofs for Kazhdan's property (T) satisfy strong duality and that the dual programs have a geometric interpretation in terms of harmonic cocycles. By dualizing geometric arguments about cocycles, we are able to simplify the property (T) SDP in the case where it carries a symmetry by finite-order inner automorphisms. As an application, we simplify the SDP proof for SL(n,ℤ) and we prove that Aut(F₄) has property (T).

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Related

• Kazhdan's Property (T) via Semidefinite Optimization
• On property (T) for Aut(F_n) and SL_n(ℤ)
• Aspects of Dualities
• An alternative proof of Kazhdan property for elementary groups
• Kazhdan's Theorem on Arithmetic Varieties