With Piotr Nowak.
Theorem 1 Let be a Banach space. Let act by homeos on compact space preserving a probability measure . Assume that has a spectral gap on some . Then
- does not coarsely embed in ,
- the -distorsion of slice is .
Theorem 2 Let be a tower of finite index subgroups whose intersection is trivial. Let be the corresponding completion. Then, for some well-chosen metric on , the box space embeds -quasi-isometrically in .
Usinf recent results of Delbie-Khukhro, we see that, depending on the choice of metric on , coarsely embeds or not in Hilbert space.
Theorem 3 If action on is measure preserving and has a spectral gap, then does not satisfy the coarse Baum-Connes conjecture.