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 my 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 ${\displaystyle asdim(X)\leq n}$ if for every ${\displaystyle R\geq 1}$ there exists a uniformly bounded cover ${\displaystyle {\mathcal {U}}}$ of $X$ such that every closed $R$-ball in $X$ intersects at most $n+1$ subsets from ${\displaystyle {\mathcal {U}}}$. Here 'uniformly bounded' means that ${\displaystyle sup_{U\in {\mathcal {U}}}diam(U)<\infty }$.

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

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

Examples

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

Properties

• If $Y\subseteq X$ is a subspace of a metric space $X$, then ${\displaystyle asdim(Y)\leq asidm(X)}$.
• For any metric spaces $X$ and $Y$ one has ${\displaystyle asdim(X\times Y)\leq asdim(X)+asdim(Y)}$.
• If $A,B\subseteq X$ then ${\displaystyle asdim(A\cup B)\leq \max\{asdim(A),asdim(B)\}}$.
• If ${\displaystyle f:Y\to X}$ is a coarse embedding (e.g. a quasi-isometric embedding), then ${\displaystyle asdim(Y)\leq asidm(X)}$.
• If $X$ and $Y$ are coarsely equivalent metric spaces (e.g. quasi-isometric metric spaces), then ${\displaystyle asdim(X)=asidm(Y)}$.
• If $X$ is a real tree then ${\displaystyle asdim(X)\leq 1}$.
• 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 ${\displaystyle \{f^{-1}(B_{r}(y))\}_{y\in Y}}$ satisfies the inequality ${\displaystyle asdim\leq n}$ uniformly. Then ${\displaystyle asdim(X)\leq asdim(Y)+n.}$ See^[3]
• If $X$ is a metric space with ${\displaystyle asdim(X)<\infty }$ then $X$ admits a coarse (uniform) embedding into a Hilbert space.^[4]
• If $X$ is a metric space of bounded geometry with ${\displaystyle asdim(X)\leq n}$ then $X$ admits a coarse embedding into a product of $n+1$ 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 $G$ 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 ${\displaystyle asdim(G)<\infty }$, then $G$ 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 $G$ is a word-hyperbolic group then ${\displaystyle asdim(G)<\infty }$.^[8]
• If $G$ is relatively hyperbolic with respect to subgroups ${\displaystyle H_{1},\dots ,H_{k}}$ each of which has finite asymptotic dimension then ${\displaystyle asdim(G)<\infty }$.^[9]
• ${\displaystyle asdim(\mathbb {Z} ^{n})=n}$.
• If $H\leq G$, where $H,G$ are finitely generated, then ${\displaystyle asdim(H)\leq asdim(G)}$.
• For Thompson's group F we have ${\displaystyle asdim(F)=\infty }$ since $F$ contains subgroups isomorphic to ${\mathbb Z}^{n}$ for arbitrarily large $n$.
• If $G$ is the fundamental group of a finite graph of groups $\mathbb {A}$ with underlying graph $A$ and finitely generated vertex groups, then^[10]

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

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

References

1. M. Gromov, Asymptotic invariants of infinite groups. Geometric group theory, Vol. 2 (Sussex, 1991), 1–295, London Math. Soc. Lecture Note Ser., 182, Cambridge Univ. Press, Cambridge, 1993; ISBN 0-521-44680-5
2. G. Yu, The Novikov conjecture for groups with finite asymptotic dimension, Annals of Mathematics 147 (1998), no. 2, 325-355.
3. G.C. Bell, A.N. Dranishnikov, A Hurewicz-type theorem for asymptotic dimension and applications to geometric group theory, Transactions of the American Mathematical Society 358 (2006), no. 11, 4749–4764.
4. John Roe, Lectures on Coarse Geometry, Univ. Lecture Ser., vol. 31, Amer. Math. Soc., Providence, RI, 2003. ISBN 0-8218-3332-4.
5. Alexander Dranishnikov, On hypersphericity of manifolds with finite asymptotic dimension. Transactions of the American Mathematical Society 355 (2003), no. 1, 155–167.
6. Alexander Dranishnikov, Asymptotic topology. (Russian) Uspekhi Mat. Nauk '55 (2000), no. 6, 71--116; translation in Russian Mathematical Surveys 55 (2000), no. 6, 1085–1129.
7. Guoliang Yu. The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space. Inventiones Mathematicae 139 (2000), no. 1, 201–240.
8. John Roe, Hyperbolic groups have finite asymptotic dimension, Proceedings of the American Mathematical Society 133 (2005), no. 9, 2489–2490
9. Densi Osin, Asymptotic dimension of relatively hyperbolic groups, International Mathematics Research Notices, 2005, no. 35, 2143–2161
10. G. Bell, and A. Dranishnikov, On asymptotic dimension of groups acting on trees. Geometriae Dedicata 103 (2004), 89–101.
11. Mladen Bestvina, and Koji Fujiwara, Bounded cohomology of subgroups of mapping class groups, Geometry & Topology 6 (2002), 69–89.
12. Lizhen Ji, Asymptotic dimension and the integral K-theoretic Novikov conjecture for arithmetic groups, Journal of Differential Geometry 68 (2004), no. 3, 535–544.
13. Karen Vogtmann, On the geometry of Outer space, Bulletin of the American Mathematical Society 52 (2015), no. 1, 27-46; Ch. 9.1

Further reading

• Gregory Bell, and Alexander Dranishnikov, Asymptotic dimension. Topology and its Applications 155 (2008), no. 12, 1265–1296.
• Sergei Buyalo, and Schroeder, Elements of asymptotic geometry. EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich, 2007. ISBN 978-3-03719-036-4