Noncommutative real algebraic geometry of Kazhdan's property (T)

Abstract

It is well-known that a finitely generated group $\Gamma$ has Kazhdan's property (T) if and only if the Laplacian element $\Delta$ in $\mathbb{R}[\Gamma ]$ has a spectral gap. In this paper, we prove that this phenomenon is witnessed in $\mathbb{R}[\Gamma ]$. Namely, $\Gamma$ has property (T) if and only if there are a constant $\kappa >0$ and a finite sequence ${\xi }_1,...,{\xi }_n$ in $\mathbb{R}[\Gamma ]$ such that ${\Delta }^2-\kappa \Delta ={\sum }_i{\xi }_i^{\ast }{\xi }_i$. This result suggests the possibility of finding new examples of property (T) groups by solving equations in $\mathbb{R}[\Gamma ]$, possibly with an assist of computers.

6 pages; a few improvement (v2); update (v3)

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