A computer-assisted proof of Kazhdan’s property (T) for automorphism groups of free groups

Abstract: Property (T) was introduced in 1967 by Kazhdan and is an important rigidity property of groups. The most elementary way to define it is through a fixed point property: a group G has property (T) if every action of G by affine isometries on a Hilbert space has a fixed point. Property (T) has numerous applications in the form of rigidity of actions and operator algebras associated to the group, constructions of expander graphs or constructions of counterexamples to Baum-Connes-type conjectures.

In this talk I will give a brief introduction to property (T) and explain the necessary group-theoretic background in order to present a computer-assisted approach to proving property (T) by showing that the Laplacian on the group has a spectral gap. This approach allowed us show that Aut(F_n), the group of automorphisms of the free group F_n on n generators, has property (T) when n is at least 5: the case n=5 is joint work with Marek Kaluba and Narutaka Ozawa, and the case of n at least 6 is joint work with Kaluba and Dawid Kielak. Important aspects of our methods include passing from a computational result to a rigorous proof and later obtaining the result for an infinite family of groups using a single computation. I will present an overview of these arguments.

analysis of PDEsclassical analysis and ODEsdynamical systemsfunctional analysisnumerical analysis

Audience: researchers in the discipline

CRM CAMP (Computer-Assisted Mathematical Proofs) in Nonlinear Analysis

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