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

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

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

Examples

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

Properties

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

Asymptotic dimension in geometric group theory

Asymptotic dimension achieved particular prominence in geometric group theory after a 1998 paper of Guoliang Yu^[2] , which proved that if ${\displaystyle G}$ is a finitely generated group of finite homotopy type (that is with a classifying space of the homotopy type of a finite CW-complex) such that ${\displaystyle \operatorname {asdim} (G)<\infty }$, then ${\displaystyle G}$ satisfies the Novikov conjecture. As was subsequently shown,^[6] finitely generated groups with finite asymptotic dimension are topologically amenable, i.e. satisfy Guoliang Yu's Property A introduced in^[7] and equivalent to the exactness of the reduced C*-algebra of the group.

• If ${\displaystyle G}$ is a word-hyperbolic group then ${\displaystyle \operatorname {asdim} (G)<\infty }$.^[8]
• If ${\displaystyle G}$ is relatively hyperbolic with respect to subgroups ${\displaystyle H_{1},\dots ,H_{k}}$ each of which has finite asymptotic dimension then ${\displaystyle \operatorname {asdim} (G)<\infty }$.^[9]
• ${\displaystyle \operatorname {asdim} (\mathbb {Z} ^{n})=n}$.
• If ${\displaystyle H\leq G}$, where ${\displaystyle H,G}$ are finitely generated, then ${\displaystyle \operatorname {asdim} (H)\leq \operatorname {asdim} (G)}$.
• For Thompson's group F we have ${\displaystyle \operatorname {asdim} (F)=\infty }$ since ${\displaystyle F}$ contains subgroups isomorphic to ${\displaystyle \mathbb {Z} ^{n}}$ for arbitrarily large ${\displaystyle n}$.
• If ${\displaystyle G}$ is the fundamental group of a finite graph of groups ${\displaystyle \mathbb {A} }$ with underlying graph ${\displaystyle A}$ and finitely generated vertex groups, then^[10]

$${\displaystyle \operatorname {asdim} (G)\leq 1+\max _{v\in VY}\operatorname {asdim} (A_{v}).}$$

• Mapping class groups of orientable finite type surfaces have finite asymptotic dimension.^[11]
• Let ${\displaystyle G}$ be a connected Lie group and let ${\displaystyle \Gamma \leq G}$ be a finitely generated discrete subgroup. Then ${\displaystyle \operatorname {asdim} (\Gamma )<\infty }$.^[12]
• It is not known if ${\displaystyle Out(F_{n})}$ has finite asymptotic dimension for ${\displaystyle n>2}$.^[13]