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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]

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 $\operatorname {asdim} (G)<\infty$ , 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 $\operatorname {asdim} (G)<\infty$ .^[8]
If $G$ is relatively hyperbolic with respect to subgroups $H_{1},\dots ,H_{k}$ each of which has finite asymptotic dimension then $\operatorname {asdim} (G)<\infty$ .^[9]
$\operatorname {asdim} (\mathbb {Z} ^{n})=n$ .
If $H\leq G$ , where $H,G$ are finitely generated, then $\operatorname {asdim} (H)\leq \operatorname {asdim} (G)$ .
For Thompson's group F we have $asdim(F)=\infty$ 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]

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

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

References

^ Gromov, Mikhael (1993). "Asymptotic Invariants of Infinite Groups". Geometric Group Theory. London Mathematical Society Lecture Note Series. Vol. 2. Cambridge University Press. ISBN 978-0-521-44680-8.
^ ^a ^b Yu, G. (1998). "The Novikov conjecture for groups with finite asymptotic dimension". Annals of Mathematics. 147 (2): 325–355. doi:10.2307/121011. JSTOR 121011. S2CID 17189763.
^ Bell, G.C.; Dranishnikov, A.N. (2006). "A Hurewicz-type theorem for asymptotic dimension and applications to geometric group theory". Transactions of the American Mathematical Society. 358 (11): 4749–64. doi:10.1090/S0002-9947-06-04088-8. MR 2231870.
^ Roe, John (2003). Lectures on Coarse Geometry. University Lecture Series. Vol. 31. American Mathematical Society. ISBN 978-0-8218-3332-2.
^ Dranishnikov, Alexander (2003). "On hypersphericity of manifolds with finite asymptotic dimension". Transactions of the American Mathematical Society. 355 (1): 155–167. doi:10.1090/S0002-9947-02-03115-X. MR 1928082.
^ Dranishnikov, Alexander (2000). "Асимптотическая топология" [Asymptotic topology]. Uspekhi Mat. Nauk (in Russian). 55 (6): 71–16. doi:10.4213/rm334.
Dranishnikov, Alexander (2000). "Asymptotic topology". Russian Mathematical Surveys. 55 (6): 1085–1129. arXiv:math/9907192. Bibcode:2000RuMaS..55.1085D. doi:10.1070/RM2000v055n06ABEH000334. S2CID 250889716.
^ Yu, Guoliang (2000). "The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space". Inventiones Mathematicae. 139 (1): 201–240. Bibcode:2000InMat.139..201Y. doi:10.1007/s002229900032. S2CID 264199937.
^ Roe, John (2005). "Hyperbolic groups have finite asymptotic dimension". Proceedings of the American Mathematical Society. 133 (9): 2489–90. doi:10.1090/S0002-9939-05-08138-4. MR 2146189.
^ Osin, Densi (2005). "Asymptotic dimension of relatively hyperbolic groups". International Mathematics Research Notices. 2005 (35): 2143–61. arXiv:math/0411585. doi:10.1155/IMRN.2005.2143. S2CID 16743152.{{cite journal}}: CS1 maint: unflagged free DOI (link)
^ Bell, G.; Dranishnikov, A. (2004). "On asymptotic dimension of groups acting on trees". Geometriae Dedicata. 103 (1): 89–101. arXiv:math/0111087. doi:10.1023/B:GEOM.0000013843.53884.77. S2CID 14631642.
^ Bestvina, Mladen; Fujiwara, Koji (2002). "Bounded cohomology of subgroups of mapping class groups". Geometry & Topology. 6 (1): 69–89. arXiv:math/0012115. doi:10.2140/gt.2002.6.69. S2CID 11350501.
^ Ji, Lizhen (2004). "Asymptotic dimension and the integral K-theoretic Novikov conjecture for arithmetic groups" (PDF). Journal of Differential Geometry. 68 (3): 535–544. doi:10.4310/jdg/1115669594.
^ Vogtmann, Karen (2015). "On the geometry of Outer space". Bulletin of the American Mathematical Society. 52 (1): 27–46. doi:10.1090/S0273-0979-2014-01466-1. MR 3286480. Ch. 9.1

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