[Paper Review] Inverse limit spaces satisfying a Poincare inequality

Jeff Cheeger, Bruce Kleiner|arXiv (Cornell University)|Dec 18, 2013

Geometric Analysis and Curvature Flows5 references4 citations

TL;DR

This paper establishes sufficient conditions for inverse limit spaces of metric measure graphs to satisfy a (1,1)-Poincaré inequality and doubling, making them PI spaces. Using a novel inductive averaging method via continuous fuzzy sections across projections, the authors prove uniform Poincaré and doubling bounds, and construct a broad class of new PI spaces, including non-bilipschitz-embeddable examples in Banach spaces with the Radon-Nikodym Property.

ABSTRACT

We give conditions on Gromov-Hausdorff convergent inverse systems of metric measure graphs (and certain higher dimensional inverse systems of metric measure spaces) 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 Poincare inequality in the sense of Heinonen-Koskela. 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. Generically our graph examples have the property that they do not bilipschitz embed in any Banach space with Radon-Nikodym property, but they do embed in the Banach space L_1. For Laakso spaces, these facts were discussed in our earlier papers.

Motivation & Objective

• To identify conditions on inverse systems of metric measure graphs that ensure the Gromov-Hausdorff limit is a PI space.
• To develop a new inductive proof technique for the (1,1)-Poincaré inequality using averaging over fibers via continuous fuzzy sections.
• To systematically construct new examples of PI spaces, including non-bilipschitz-embeddable ones in Banach spaces with the Radon-Nikodym Property.
• To extend the framework to higher-dimensional inverse systems using cube complexes and verify the Poincaré inequality via path families and fuzzy sections.

Proposed method

• Define 'admissible' inverse systems via six axioms ensuring bounded local geometry, measure compatibility, and continuity across codimension-1 faces.
• Use continuous fuzzy sections—probability measures on fibers of projection maps—that vary continuously in the weak topology.
• Prove the (1,1)-Poincaré inequality inductively by averaging functions over fibers using the fuzzy sections and upper gradients.
• Construct examples inductively by making local, independent choices on each level of the inverse system.
• Apply a second proof strategy using a natural probability measure on path lifts to verify the Poincaré inequality.
• Extend the framework to higher-dimensional systems using cube complexes with controlled gallery diameter and measure compatibility.

Experimental results

Research questions

• RQ1 Under what conditions on inverse systems of metric measure graphs does the Gromov-Hausdorff limit satisfy a (1,1)-Poincaré inequality and doubling?
• RQ2 Can a novel inductive averaging method using continuous fuzzy sections be used to prove the Poincaré inequality in such limits?
• RQ3 What classes of new PI spaces can be systematically constructed using this framework, and what are their geometric and analytic properties?
• RQ4 Do these constructed PI spaces fail to bilipschitz embed in Banach spaces with the Radon-Nikodym Property, and why?
• RQ5 Can the framework be extended to higher-dimensional inverse systems, and what modifications are needed for the Poincaré inequality to hold?

Key findings

• The measured Gromov-Hausdorff limit of an admissible inverse system is a PI space satisfying a (1,1)-Poincaré inequality.
• The doubling constant and Poincaré inequality constants depend only on the system's structural parameters: m ≥ 2, Δ, θ, C.
• The inverse limit has topological dimension 1, analytic dimension 1, and Hausdorff dimension >1 except in degenerate cases.
• Generically, these spaces do not bilipschitz embed in any Banach space with the Radon-Nikodym Property, extending a result for Laakso spaces.
• The Poincaré inequality can be proven via two distinct methods: continuous fuzzy sections and path family measures.
• The framework generalizes to higher-dimensional inverse systems using cube complexes, with the Poincaré inequality preserved under appropriate axioms.