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Kazhdan’s Property (T) via Semidefinite Optimization

Tim Netzer
Andreas Thom
Journal / Anthology

Experimental Mathematics
Year: 2015
Volume: 24
Issue: 3
Page range: 371–374

Following an idea of Ozawa, we give a new proof of Kazhdan’s property (T) for SL(3, Z) by showing that 2 − /6 is a Hermitian sum of squares in the group algebra, whereis the unnormalized Laplace operator with respect to the natural generating set. This corresponds to a spectral gap of 1/72 ≈ 0.014 for the associated random-walk operator. The sum-of-squares representation was found numerically by a semidefinite programming algorithm and then turned into an exact symbolic representation, provided in an attached Mathematica file.