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According to the wikipedia page on "Isoperimetric dimension", the isoperimetric dimension is invariant under quasi-isometries, even between manifolds and graphs:

"[...] the isoperimetric dimension is preserved by quasi isometries, both by quasi-isometries between manifolds, between graphs, and even by quasi isometries carrying manifolds to graphs, with the respective definitions."

Does someone know the precise definitions of this, and references on where to find this kind of result?

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This should become trivial once you make right definition. –  ε-δ Jan 9 '12 at 20:33

2 Answers 2

up vote 4 down vote accepted

You can find definitions and properties in Fan Chung's paper, "Discrete Isoperimetric Inequalities," Surveys in Differential Geometry IX, International Press, 2004, 53--82 (PDF download link). She says, "In a way, a graph can be viewed as a discretization of a Riemannian manifold in $\mathbb{R}^n$ where $n$ is roughly equal to [the isoperimetric dimension] $\delta$." Here is the precise definition of $\delta$:

We say that a graph $G$ has isoperimetric dimension $\delta$ with an isoperimetric constant $c_\delta$ if for all subsets $X$ of $V(G)$, the number of edges between $X$ and the complement $\bar{X}$ of $X$, denoted by $e(X,\bar{X})$, satisifies $$e(X,\bar{X}) \ge c_\delta \; \mathrm{vol} (X)^{\frac{\delta-1}{\delta}}$$ where $\mathrm{vol} (X) \le \mathrm{vol} (\bar{X})$ and $c_\delta$ is a constant depending only on $\delta$.

Earlier:

For a graph $G$ and a subset $X$ of vertices in $G$, the volume vol$(X)$ is defined by $$\mathrm{vol} (X) = \sum_{v \in X} d_v$$ where $d_v$ is the degree of $v$.

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A good ref. for this is:

"Isoperimetric inequalities: differential geometric and analytic perspectives" Isaac Chavel

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