Laminations hyperfinies et revetements
  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. 
M. Berm\'udez and G. Hector