Asymptotic dimension

In metric geometry, asymptotic dimension of a metric space is a large-scale analog of Lebesgue covering dimension. The notion of asymptotic dimension was introduced by Mikhail Gromov in his 1993 monograph Asymptotic invariants of infinite groups^[1] in the context of geometric group theory, as a quasi-isometry invariant of finitely generated groups. As shown by Guoliang Yu, finitely generated groups of finite homotopy type with finite asymptotic dimension satisfy the Novikov conjecture.^[2] Asymptotic dimension has important applications in geometric analysis and index theory.

Formal definition

Let $X$ be a metric space and $n\geq 0$ be an integer. We say that $\operatorname {asdim} (X)\leq n$ if for every $R\geq 1$ there exists a uniformly bounded cover ${\mathcal {U}}$ of $X$ such that every closed $R$ -ball in $X$ intersects at most $n+1$ subsets from ${\mathcal {U}}$ . Here 'uniformly bounded' means that $\sup _{U\in {\mathcal {U}}}\operatorname {diam} (U)<\infty$ .

We then define the asymptotic dimension $\operatorname {asdim} (X)$ as the smallest integer $n\geq 0$ such that $\operatorname {asdim} (X)\leq n$ , if at least one such $n$ exists, and define $\operatorname {asdim} (X):=\infty$ otherwise.

Also, one says that a family $(X_{i})_{i\in I}$ of metric spaces satisfies $\operatorname {asdim} (X)\leq n$ uniformly if for every $R\geq 1$ and every $i\in I$ there exists a cover ${\mathcal {U}}_{i}$ of $X_{i}$ by sets of diameter at most $D(R)<\infty$ (independent of $i$ ) such that every closed $R$ -ball in $X_{i}$ intersects at most $n+1$ subsets from ${\mathcal {U}}_{i}$ .

Examples

If $X$ is a metric space of bounded diameter then $\operatorname {asdim} (X)=0$ .
$\operatorname {asdim} (\mathbb {R} )=\operatorname {asdim} (\mathbb {Z} )=1$ .
$\operatorname {asdim} (\mathbb {R} ^{n})=n$ .
$\operatorname {asdim} (\mathbb {H} ^{n})=n$ .

Properties

If $Y\subseteq X$ is a subspace of a metric space $X$ , then $\operatorname {asdim} (Y)\leq \operatorname {asdim} (X)$ .
For any metric spaces $X$ and $Y$ one has $\operatorname {asdim} (X\times Y)\leq \operatorname {asdim} (X)+\operatorname {asdim} (Y)$ .
If $A,B\subseteq X$ then $\operatorname {asdim} (A\cup B)\leq \max\{\operatorname {asdim} (A),\operatorname {asdim} (B)\}$ .
If $f:Y\to X$ is a coarse embedding (e.g. a quasi-isometric embedding), then $\operatorname {asdim} (Y)\leq \operatorname {asdim} (X)$ .
If $X$ and $Y$ are coarsely equivalent metric spaces (e.g. quasi-isometric metric spaces), then $\operatorname {asdim} (X)=\operatorname {asdim} (Y)$ .
If $X$ is a real tree then $\operatorname {asdim} (X)\leq 1$ .
Let $f:X\to Y$ be a Lipschitz map from a geodesic metric space $X$ to a metric space $Y$ . Suppose that for every $r>0$ the set family $\{f^{-1}(B_{r}(y))\}_{y\in Y}$ satisfies the inequality $\operatorname {asdim} \leq n$ uniformly. Then $\operatorname {asdim} (X)\leq \operatorname {asdim} (Y)+n.$ See^[3]
If $X$ is a metric space with $\operatorname {asdim} (X)<\infty$ then $X$ admits a coarse (uniform) embedding into a Hilbert space.^[4]
If $X$ is a metric space of bounded geometry with $\operatorname {asdim} (X)\leq n$ then $X$ admits a coarse embedding into a product of $n+1$ locally finite simplicial trees.^[5]