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This week there is a conference here at IPAM on expanders in pure and applied mathematics. I was an invited speaker, but I don’t actually work in expanders per se (though I am certainly interested in them). So I spoke instead about the recent simplified proof by Kleiner of the celebrated theorem of Gromov on groups of polynomial growth. (This proof does not directly mention expanders, but the argument nevertheless hinges on the absence of expansion in the Cayley graph of a group of polynomial growth, which is exhibited through the smoothness properties of harmonic functions on such graphs.)

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