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Non-embeddability of Carnot Groups into L^1

Lisa Naples, Macalester College
Tuesday, May 9, 2023 - 1:30pm to 3:30pm
PDL C-401

In 2006, Cheeger and Kleiner proved that the Heisenberg group fails to bi-Lipschitzly embed into \$L^1\$. The authors remarked that their proof should hold for every Carnot group \$G\$ that satisfies: for every finite perimeter subset \$E\subset G\$ and \$\operatorname{Per}_E\$-almost every , every tangent of \$E\$ at \$x\$ is a half-space. While such property remains elusive, Ambrosio, Kleiner, and Le Donne made significant progress by proving that every Carnot group  satisfies: every finite perimeter subset has the property that for \$\operatorname{Per_E}\$-almost every \$x\in G\$ , there is some blowup of \$E\$ at \$x\$ that is a half-space. We use the result of Ambrosio, Kleiner, and Le Donne, together with the framework of Cheeger and Kleiner to show that nonabelian Carnot groups fail to bilischitz embed into \$L^1\$. This is joint work with Sylvester Eriksson-Bique, Chris Gartland, Enrico Le Donne, and Sebastiano Nicolussi-Golo.

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