Computer proofs for Property (T), and SDP duality

Martin Nitsche

Computer proofs for Property (T), and SDP duality

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arxiv 2009.05134 v3 pith:BDH5DBFJ submitted 2020-09-10 math.GR math.OA

Computer proofs for Property (T), and SDP duality

Martin Nitsche This is my paper

classification math.GR math.OA

keywords propertycocyclescomputerdualitygeometricprogramsproofssimplify

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

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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 $SL(n,\mathbb{Z})$ and we prove that $Aut(F_4)$ has property (T).

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Cited by 2 Pith papers

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Invariant trace simplices and relative property (T)
math.OA 2026-04 unverdicted novelty 7.0

If (G,H) has relative property (T) and H-actions on von Neumann algebras of extremal invariant traces are ergodic, then the simplex T(A)^G of G-invariant traces is Bauer.
Abelianizations of finite-index subgroups of the handlebody group
math.GT 2026-07 accept novelty 6.0

For genus ≥ 4, meridian multitwists vanish in H_1 of any finite-index subgroup of the handlebody group, and subgroups containing the Torelli group, twist group, or Johnson kernel have trivial rational abelianization.