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.


About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See
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