## Notes of Damian Sawicki’s Southampton lecture 28-03-2017

Warped cones

With Piotr Nowak.

Theorem 1 Let ${E}$ be a Banach space. Let ${G}$ act by homeos on compact space ${Y}$ preserving a probability measure ${\mu}$. Assume that ${G}$ has a spectral gap on some ${L^p(Y,\mu;E)}$. Then

1. ${O_G(Y)}$ does not coarsely embed in ${E}$,
2. the ${E}$-distorsion of slice ${\{t\}\times Y}$ is ${\Theta(\log t)}$.

Theorem 2 Let ${G_i}$ be a tower of finite index subgroups whose intersection is trivial. Let ${\hat G}$ be the corresponding completion. Then, for some well-chosen metric on ${\hat G}$, the box space embeds ${(1,1)}$-quasi-isometrically in ${O_G(\hat G)}$.

Usinf recent results of Delbie-Khukhro, we see that, depending on the choice of metric on ${\hat G}$, ${O_G(\hat G)}$ coarsely embeds or not in Hilbert space.

Theorem 3 If ${G}$ action on ${Y}$ is measure preserving and has a spectral gap, then ${O_G(Y)}$ does not satisfy the coarse Baum-Connes conjecture.