  We give sufficient conditions for a metric space to bilipschitz embed in L_1.
In particular, if X is a length space and there is a Lipschitz map u:X--->R
such that for every interval I in R, the connected components of the inverse
image f^{-1}(I) have diameter at most a constant time the diameter of I, then X
admits a bilipschitz embedding in L_1. As a corollary, well-known examples of
Laakso bilipschitz embed in L_1, though they do not embed in any any Banach
space with the Radon-Nikodym property (e.g. the space l_1 of summable
sequences).
  The spaces appearing the statement of the bilipschitz embedding theorem have
an alternate characterization as inverse limits of systems of metric graphs
satisfying certain additional conditions. This representation, which may be of
independent interest, is the initial part of the proof of the bilipschitz
embedding theorem. The rest of the proof uses the combinatorial structure of
the inverse system of graphs and a diffusion construction, to produce the
embedding in L_1.
