[Paper Review] Expansion properties of metric spaces not admitting a coarse embedding into a Hilbert space

Mikhail I. Ostrovskii|ArXiv.org|Mar 3, 2009

Advanced Operator Algebra Research7 references3 citations

TL;DR

This paper establishes a weak expansion property in locally finite metric spaces that do not coarsely embed into Hilbert space, showing that such spaces contain subgraphs with controlled vertex boundary growth under a weighted measure. The key result is a quantitative lower bound on the expansion-like behavior of these subgraphs, which supports the conjecture that non-embeddable spaces contain weak expanders.

ABSTRACT

The main purpose of the paper is to find some expansion properties of locally finite metric spaces which do not embed coarsely into a Hilbert space. The obtained result is used to show that infinite locally finite graphs excluding a minor embed coarsely into a Hilbert space. In an appendix a direct proof of the latter result is given.

Motivation & Objective

• To identify expansion-like properties in locally finite metric spaces that do not admit coarse embeddings into Hilbert space.
• To investigate whether such spaces contain weak expander families, a long-standing open problem in geometric group theory and metric geometry.
• To provide a quantitative expansion condition for subgraphs derived from sets violating coarse embeddability into $L_1$.
• To support the conjecture that non-coarsely embeddable spaces with bounded geometry contain weak expanders by establishing a weaker expansion property.
• To offer a direct proof in an appendix for the coarse embeddability of infinite locally finite graphs excluding a minor into Hilbert space.

Proposed method

• Constructs a sequence of finite subsets $M_n \subset M$ and a probability measure $\mu_n$ on $M_n \times M_n$ satisfying a uniform bound on the $L_1$-norm of Lipschitz functions.
• Defines graphs $G(n,s)$ with edges between vertices within distance $s$, using the metric $d_M$ to control connectivity.
• Introduces a weighted measure $\nu_n$ on $M_n$ derived from $\mu_n$, and analyzes vertex boundaries $\delta_F A$ in induced subgraphs $F$.
• Uses an iterative exhaustion process to remove low-boundary sets, assuming no good subgraph exists to derive a contradiction.
• Constructs a family of $1$-Lipschitz functions $f_\theta$ indexed by $\theta \in \{-1,1\}^p$, and embeds them into $L_1$ via averaging over a probability space.
• Applies a result from [KPR93] to decompose graphs into components with bounded diameter, and uses this to construct a coarse embedding into $L_1$ via a weighted sum of $L_1$-valued functions.

Experimental results

Research questions

• RQ1 Do locally finite metric spaces not coarsely embeddable into Hilbert space exhibit any form of expansion in their finite subsets?
• RQ2 Can a weak expansion property be established for graphs derived from such spaces, even if full expanders are not present?
• RQ3 Is there a quantitative lower bound on the vertex boundary growth in induced subgraphs under a weighted measure?
• RQ4 Does the absence of coarse embeddability into $L_1$ imply the existence of a structural expansion-like feature in the space?
• RQ5 Can the coarse embeddability of infinite locally finite graphs excluding a minor into Hilbert space be proven directly via this framework?

Key findings

• For $s > 8D$ and $2n > s$, the graph $G(n,s)$ contains an induced subgraph $F$ with $d_M$-diameter at least $n - s/2$ such that every subset $A \subset F$ of smaller $d_M$-diameter satisfies $\nu_n(\delta_F A) > \varphi(D,s)\nu_n(A)$, where $\varphi(D,s) = \frac{s}{4D} - 2$.
• The expansion-like condition is uniform in $n$, with the expansion factor $\varphi(D,s)$ depending only on $D$ and $s$, and growing linearly with $s$.
• The construction leads to a coarse embedding of the space into $L_1$, with the embedding norm satisfying $||\varphi(u) - \varphi(v)||_{L_1} \leq 3d(u,v)$ and $||\varphi(u) - \varphi(v)||_{L_1} \geq (4/3)^i \varepsilon_r$ when $d(u,v) \geq d_{2^i, r}$.
• The proof establishes that infinite locally finite graphs excluding a minor embed coarsely into Hilbert space, via a direct construction in the appendix.
• The result supports the conjecture that non-coarsely embeddable spaces with bounded geometry contain weak expanders, by proving a weaker but quantitatively controlled expansion property.
• The key inequality $\int_{\Omega_\Delta} |F_{\Delta,u}| d\omega \geq \varepsilon_r \Delta$ ensures that the $L_1$-norm of the embedding grows with $\Delta$, enabling the coarse embedding condition.