[Paper Review] The asymptotic rank of metric spaces

Stefan Wenger|ArXiv.org|Jan 8, 2007

Geometric Analysis and Curvature Flows18 references4 citations

TL;DR

This paper introduces the asymptotic rank of metric spaces as a geometric invariant that generalizes Euclidean rank, characterizing it via the growth rate of higher filling functions. The main result shows that for complete metric spaces with cone-type inequalities up to dimension k, if the asymptotic rank is at most k, then the (k+1)st filling volume function grows sub-Euclideanly, implying a sub-Euclidean isoperimetric inequality.

ABSTRACT

In this article we define and study a notion of asymptotic rank for metric spaces and show in our main theorem that for a large class of spaces, the asymptotic rank is characterized by the growth of the higher filling functions. For a proper, cocompact, simply-connected geodesic metric space of non-curvature in the sense of Alexandrov the asymptotic rank equals its Euclidean rank.

Motivation & Objective

• To define and study a new geometric invariant, the asymptotic rank, for general metric spaces.
• To generalize the concept of Euclidean rank beyond non-positively curved spaces.
• To characterize asymptotic rank in terms of the asymptotic behavior of higher filling functions.
• To establish a connection between geometric structure and isoperimetric growth in complete metric spaces with controlled geometry.
• To extend known results on filling functions in Hadamard spaces and symmetric spaces of non-compact type.

Proposed method

• Define asymptotic rank using bi-Lipschitz embeddings of positive measure compact subsets of R^n into asymptotic subsets of the space.
• Use integral currents to formalize k-chains and k-cycles, enabling the definition of filling volume functions FV_{k+1}(r).
• Introduce cone-type inequalities and generalized combings with length and distortion functions to control filling volumes.
• Apply metric differentiation and area formulas to bound the mass of filling currents via Lipschitz contractions.
• Establish diameter-volume inequalities for currents using generalized combings and metric differentiation techniques.
• Prove that sub-Euclidean growth of FV_{k+1}(r) follows when the asymptotic rank is ≤ k and cone-type inequalities hold up to dimension k.

Experimental results

Research questions

• RQ1 How can the Euclidean rank of a metric space be generalized to non-positively curved or general metric spaces?
• RQ2 What is the relationship between the asymptotic rank of a space and the growth rate of its higher filling functions?
• RQ3 Under what geometric conditions does a metric space admit a sub-Euclidean isoperimetric inequality for k-cycles?
• RQ4 Can the asymptotic rank be characterized via the existence of asymptotic cones with certain dimensional properties?
• RQ5 To what extent do cone-type inequalities imply sub-Euclidean filling volume growth?

Key findings

• The asymptotic rank of a proper, cocompact, simply-connected geodesic metric space of non-positive curvature equals its Euclidean rank.
• For a complete metric space with a uniform chain condition (Q-bounded chains), if the asymptotic rank is at most k, then the (k+1)th filling volume function satisfies limsup_{r→∞} FV_{k+1}(r)/r^{(k+1)/k} = 0.
• Spaces admitting cone-type inequalities for all m ≤ k also satisfy sub-Euclidean isoperimetric inequalities for k-cycles.
• The result extends to Hadamard spaces and symmetric spaces of non-compact type, where stronger linear filling inequalities are known above the rank.
• Generalized combings with polynomial length and distortion functions imply diameter-volume inequalities of type (ν + kμ, 1), leading to cone-type inequalities when ν=1 and μ=0.
• The proof relies on metric differentiation, area formulas, and bounds on Jacobians of Lipschitz contractions to control the mass of filling currents.