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{{cite book |first=Mikhael |last=Gromov |chapter=Asymptotic Invariants of Infinite Groups |title=Geometric Group Theory |url=https://books.google.com/books?id=dH02YAfVqkYC |date=1993 |publisher=Cambridge University Press |isbn=978-0-521-44680-8 |volume=2 |series=London Mathematical Society Lecture Note Series |issue=182}} 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. Asymptotic dimension has important applications in geometric analysis and index theory.

Formal definition

Let $X$ be a metric space and $n\ge 0$ be an integer. We say that $\operatorname{asdim}(X)\le n$ if for every $R\ge 1$ 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 $n\ge 0$ such that $\operatorname{asdim}(X)\le n$ , 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 $\operatorname{asdim}(X)\le n$ uniformly if for every $R\ge 1$ 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 $\operatorname{asdim}(Y)\le \operatorname{asdim}(X)$ .
For any metric spaces $X$ and $Y$ one has $\operatorname{asdim}(X\times Y)\le \operatorname{asdim}(X)+\operatorname{asdim}(Y)$ .
If $A,B\subseteq X$ then $\operatorname{asdim}(A\cup B)\le \max\{\operatorname{asdim}(A), \operatorname{asdim}(B)\}$ .
If $f:Y\to X$ is a coarse embedding (e.g. a quasi-isometric embedding), then $\operatorname{asdim}(Y)\le \operatorname{asdim}(X)$ .
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 $\operatorname{asdim}(X)\le 1$ .
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 $\operatorname{asdim} \le n$ uniformly. Then $\operatorname{asdim}(X)\le \operatorname{asdim}(Y) +n.$ See{{cite journal |first1=G.C. |last1=Bell |first2=A.N. |last2=Dranishnikov |title=A Hurewicz-type theorem for asymptotic dimension and applications to geometric group theory |journal=Transactions of the American Mathematical Society |volume=358 |issue=11 |pages=4749–64 |year=2006 |doi=10.1090/S0002-9947-06-04088-8 |mr=2231870|doi-access=free }}
If $X$ is a metric space with $\operatorname{asdim}(X)<\infty$ then $X$ admits a coarse (uniform) embedding into a Hilbert space.{{cite book |first=John |last=Roe |title=Lectures on Coarse Geometry |url=https://books.google.com/books?id=jbsFCAAAQBAJ |year=2003 |publisher=American Mathematical Society |isbn=978-0-8218-3332-2 |series=University Lecture Series |volume=31}}
If $X$ is a metric space of bounded geometry with $\operatorname{asdim}(X)\le n$ then $X$ admits a coarse embedding into a product of $n+1$ locally finite simplicial trees.{{cite journal |first=Alexander |last=Dranishnikov |title=On hypersphericity of manifolds with finite asymptotic dimension |journal=Transactions of the American Mathematical Society |volume=355 |issue=1 |pages=155–167 |year=2003 |doi=10.1090/S0002-9947-02-03115-X |mr=1928082|doi-access=free }}

Asymptotic dimension in geometric group theory

Asymptotic dimension achieved particular prominence in geometric group theory after a 1998 paper of Guoliang Yu{{cite journal |first=G. |last=Yu |s2cid=17189763 |title=The Novikov conjecture for groups with finite asymptotic dimension |journal=Annals of Mathematics |volume=147 |issue=2 |pages=325–355 |year=1998 |jstor=121011|doi=10.2307/121011 }}

, 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,{{cite journal |first=Alexander |last=Dranishnikov |title=Асимптотическая топология |trans-title=Asymptotic topology |journal=Uspekhi Mat. Nauk |volume=55 |issue=6 |pages=71–16 |year=2000 |doi=10.4213/rm334 |language=Russian|doi-access=free }}
{{cite journal |first=Alexander |last=Dranishnikov |title=Asymptotic topology |journal=Russian Mathematical Surveys |volume=55 |issue=6 |pages=1085–1129 |year=2000 |doi=10.1070/RM2000v055n06ABEH000334 |arxiv=math/9907192|bibcode=2000RuMaS..55.1085D |s2cid=250889716 }} finitely generated groups with finite asymptotic dimension are topologically amenable, i.e. satisfy Guoliang Yu's Property A introduced in{{cite journal |first=Guoliang |last=Yu |title=The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space |journal=Inventiones Mathematicae |volume=139 |issue=1 |pages=201–240 |year=2000 |doi=10.1007/s002229900032 |bibcode=2000InMat.139..201Y |s2cid=264199937 |citeseerx=10.1.1.155.1500 }} and equivalent to the exactness of the reduced C*-algebra of the group.

If $G$ is a word-hyperbolic group then $\operatorname{asdim}(G)<\infty$ .{{cite journal |first=John |last=Roe |title=Hyperbolic groups have finite asymptotic dimension |journal=Proceedings of the American Mathematical Society |volume=133 |issue=9 |pages=2489–90 |year=2005 |doi=10.1090/S0002-9939-05-08138-4 |mr=2146189|doi-access=free }}
If $G$ is relatively hyperbolic with respect to subgroups $H_1,\dots, H_k$ each of which has finite asymptotic dimension then $\operatorname{asdim}(G)<\infty$ .{{cite journal |first=Densi |last=Osin |title=Asymptotic dimension of relatively hyperbolic groups |journal=International Mathematics Research Notices |volume=2005 |issue=35 |pages=2143–61 |year=2005 |doi=10.1155/IMRN.2005.2143 |arxiv=math/0411585 |doi-access=|s2cid=16743152 }}
$\operatorname{asdim}(\mathbb Z^n)=n$ .
If $H\le G$ , where $H,G$ are finitely generated, then $\operatorname{asdim}(H)\le \operatorname{asdim}(G)$ .
For Thompson's group F we have $\operatorname{asdim}(F)=\infty$ 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{{cite journal |first1=G. |last1=Bell |first2=A. |last2=Dranishnikov |title=On asymptotic dimension of groups acting on trees |journal=Geometriae Dedicata |volume=103 |issue=1 |pages=89–101 |year=2004 |doi=10.1023/B:GEOM.0000013843.53884.77 |arxiv=math/0111087|s2cid=14631642 }}

$\operatorname{asdim}(G)\le 1+ \max_{v\in VY} \operatorname{asdim} (A_v).$

Mapping class groups of orientable finite type surfaces have finite asymptotic dimension.{{cite journal |first1=Mladen |last1=Bestvina |first2=Koji |last2=Fujiwara |title=Bounded cohomology of subgroups of mapping class groups |journal=Geometry & Topology |volume=6 |pages=69–89 |year=2002 |issue=1 | arxiv=math/0012115|doi=10.2140/gt.2002.6.69 |s2cid=11350501 }}
Let $G$ be a connected Lie group and let $\Gamma\le G$ be a finitely generated discrete subgroup. Then $\operatorname{asdim}(\Gamma)<\infty$ .{{cite journal |first=Lizhen |last=Ji |title=Asymptotic dimension and the integral K-theoretic Novikov conjecture for arithmetic groups |journal=Journal of Differential Geometry |volume=68 |issue=3 |pages=535–544 |year=2004 |doi= 10.4310/jdg/1115669594|doi-access=free |url=https://projecteuclid.org/journals/journal-of-differential-geometry/volume-68/issue-3/Asymptotic-dimension-and-the-integral-K-theoretic-Novikov-conjecture-for/10.4310/jdg/1115669594.pdf }}
It is not known if $Out(F_n)$ has finite asymptotic dimension for $n>2$ .{{cite journal |first=Karen |last=Vogtmann |title=On the geometry of Outer space |journal=Bulletin of the American Mathematical Society |volume=52 |issue=1 |pages=27–46 |year=2015 |doi=10.1090/S0273-0979-2014-01466-1 |mr=3286480|doi-access=free }} Ch. 9.1

asymptotic dimension

Formal definition

Examples

Properties

Asymptotic dimension in geometric group theory

References

Further reading