Abstract
We give conditions on Gromov-Hausdorff convergent inverse systems of metric measure graphs
which imply that the measured Gromov-Hausdorff limit (equivalently, the inverse limit) is a PI space i.e., it
satisfies a doubling condition and a Poincaré inequality in the sense of Heinonen-Koskela [12]. The Poincaré
inequality is actually of type (1, 1). We also give a systematic construction of examples for which our conditions
are satisfied. Included are known examples of PI spaces, such as Laakso spaces, and a large class of new
examples. As follows easily from [4], generically our examples have the property that they do not bilipschitz
embed in any Banach space with Radon-Nikodym property. For Laakso spaces, thiswas noted in [4]. However
according to [7] these spaces admit a bilipschitz embedding in L1. For Laakso spaces, this was announced
in [5].