Laminations hyperfinies et revetements

Details Laminations hyperfinies et revetements Laminations hyperfinies et revetements

2005-03-17

arxiv.org/abs/math/0503350v2

Mathematics

Dynamical Systems

32 pages, 1 figure

Scientific paper

We introduce the so-called BT-category of borelian-topological spaces: it will be a natural frame for a measurable classification of usual foliations and laminations. We focus on the two-dimensional case: borelian laminations by surfaces. We prove two main results: (1) Any borelian lamination by planes is the suspension of a $\Z^2$-action on a Borel space iff this lamination is hyperfinite. (2) Any borelian lamination by surfaces is amenable if parabolic, i.e. if it admits a complex structure parabolic on each leaf. The third result is an improvement of (2) in case of laminations endowed with a transverse quasi-invariant measure $\mu$. The statement is the following: (3) Any borelian lamination by planes, cylinders and tori is $\mu$-amenable if and only if it admits a metric which is flat on $\mu$-almost all leafs.

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