Aut(F_5) has the property (T)

Narutaka Ozawa
Kyoto University
Research Institute for Mathematical Sciences

Kazhdan's property (T) for groups has a number of applications in pure and applied mathematics. It has long been thought that groups with property (T) are rare among the "naturally-occurring" groups, but it may not be so and it may be possible to observe this by extensive computer calculations. After an introduction, I will present a computer assisted (but mathematically rigorous) method of confirming property (T) based on semidefinite programming with some operator algebraic input. I will report the progress recently made in collaboration with M. Kaluba, P. Nowak, and PL-grid, a Polish supercomputer. It confirms property (T) of $textrm{Aut}(mathbb{F}_5)$, which solves a well-known problem in geometric group theory, at least partially, leaving the tantalizing question in the case of $textrm{Aut}(mathbb{F}_d)$, $d=4$ and $d>5$, unsettled.

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