Noncommutative Real Algebraic Geometry of Kazhdan’s Property (T)

It is well known that a finitely generated group Γ has Kazhdan’s property (T) if and only if the Laplacian element Δ in R[Γ] has a spectral gap. In this paper, we prove that this phenomenon is witnessed in R[Γ] . Namely, Γ has property (T) if and only if there exist a constant κ>0 and a finite sequence ξ₁,… ,ξₙ in R[Γ] such that Δ²-κΔ=∑ ᵢξᵢξᵢ . This result suggests the possibility of finding new examples of property (T) groups by solving equations in R[Γ] , possibly with the assistance of computers.