[Paper Review] On the separation profile of infinite graphs

Itaï Benjamini, Oded Schramm+1 more|arXiv (Cornell University)|Apr 6, 2010

Geometric and Algebraic Topology15 references3 citations

TL;DR

This paper introduces the separation profile as a coarse geometric invariant for infinite graphs, particularly focusing on Cayley graphs and hyperbolic spaces. It establishes that finite separation corresponds to bounded treewidth, proves separation functions for various groups (e.g., $n / \log n$ for product of free groups, $n^{(d-1)/d}$ for $\mathbb{Z}^d$), and shows no semi-regular map exists from $\mathbb{Z}^2$ to the lamplighter group over $\mathbb{Z}$, linking separation to asymptotic dimension and harmonic functions.

ABSTRACT

Initial steps in the study of inner expansion properties of infinite Cayley graphs and other infinite graphs, such as hyperbolic ones, are taken, in a flavor similar to the well-known Lipton-Tarjan square root separation result for planar graphs. Connections to relaxed versions of quasi-isometries are explored, such as regular and semiregular maps.

Motivation & Objective

• To define and analyze the separation profile as a coarse geometric invariant for infinite graphs and path metric spaces.
• To determine the separation functions of key infinite graphs, including Cayley graphs of groups with intermediate growth, hyperbolic spaces, and products of graphs.
• To explore connections between separation profiles, regular and semi-regular maps, and invariants like asymptotic dimension and harmonic functions.
• To investigate whether linear or $n/\log n$ separation is the maximal possible growth for vertex-transitive graphs.
• To resolve structural questions about the existence of semi-regular maps between specific spaces, such as $\mathbb{Z}^2$ and the lamplighter group over $\mathbb{Z}$.

Proposed method

• Introduces the separation function $\text{sep}_G(x)$ as the supremum over all subgraphs of size $x$ of the minimum vertex cut needed to split them into components of size at most $x/2$.
• Uses the concept of regular and semi-regular maps to relate separation profiles across different spaces, showing that separation is monotone non-decreasing under such maps.
• Applies the notion of quasi-level sets and component growth in $\mathbb{Z}^2$ to prove non-existence of semi-regular maps to $\mathbb{Z}$ and the lamplighter group.
• Leverages asymptotic dimension theory, showing that if $X \to_{\text{s-reg}} Y$, then $\text{asdim}(X) \leq \text{asdim}(Y)$, and uses this to rule out certain maps.
• Analyzes specific examples: $\mathbb{Z}^d$ has separation $n^{(d-1)/d}$, $T \times T$ has separation $n / \log n$, and hyperbolic graphs have separation either constant or at least logarithmic.
• Proves that finite separation implies bounded treewidth, and characterizes the structure of graphs with finite separation.

Experimental results

Research questions

• RQ1 What separation functions are realizable by transitive or Cayley graphs, and which growth rates are possible for such graphs?
• RQ2 Is $n/\log n$ the maximal possible separation growth for vertex-transitive graphs, or can linear separation occur?
• RQ3 Does a spectral radius less than 1 combined with a weak separation condition imply the existence of non-constant bounded harmonic functions?
• RQ4 Can $\mathbb{Z}^2$ semi-regularly map into the lamplighter group over $\mathbb{Z}$, and what does this imply about asymptotic dimension and separation profiles?
• RQ5 Is there a semi-regular map from $\mathbb{H}^2$ to $T \times \mathbb{Z}$, and how does this relate to coarse geometric invariants?

Key findings

• The separation function for $\mathbb{Z}^d$ is $n^{(d-1)/d}$, showing a power-law dependence on dimension.
• The product of two regular trees $T \times T$ has separation function $\asymp n / \log n$, indicating a sharp threshold between polynomial and logarithmic growth.
• Finite separation in a graph is equivalent to bounded treewidth, providing a structural characterization.
• There is no semi-regular map from $\mathbb{Z}^2$ to $\mathbb{Z}$, and hence no such map to the lamplighter group over $\mathbb{Z}$, due to unbounded component growth in quasi-level sets.
• For Gromov-hyperbolic graphs, the separation function is either constant or grows at least logarithmically, establishing a gap theorem.
• Asymptotic dimension is monotone under semi-regular maps, and this property rules out certain maps (e.g., from $\mathbb{Z}^n$ to $\mathbb{Z}^m$ for $n > m$), providing a topological obstruction.