Realization of Metric Spaces as Inverse Limits, and Bilipschitz Embedding in L1

We give sufficient conditions for a metric space to bilipschitz embed in L1. In particular, if X is a length space and there is a Lipschitz map $${u: X \rightarrow \mathbb R}$$ such that for every interval $${I \subset \mathbb R}$$, the connected components of u−1(I) have diameter $${\leq {\rm const} \cdot {\rm diam}(I)}$$, then X admits a bilipschitz embedding in L1. As a corollary, the Laakso examples, (Geom Funct Anal 10(1):111–123, 2000), bilipschitz embed in L1, though they do not embed in any any Banach space with the Radon–Nikodym property (e.g. the space ℓ1 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 L1.