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Uniform Spectral Gaps and Paradoxical Decompositions: the Affine Plane

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If you have a question about this talk, please contact Aled Walker .

It is well-known that the non-abelian free group on two generators is paradoxical using 4 pieces. More generally, for a given space X, an action of this group on X may be paradoxical. In this talk, we will explain how paradoxical decompositions are directly related to spectral gaps for related unitary representations, and how the number of pieces required in the paradoxical decompositions relates to the actual value of the spectral gap. We will present some results on how to find paradoxical decompositions of the plane using affine transformations along with some applications to Kazhdan’s Property (T).

This talk is part of the Discrete Analysis Seminar series.

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