Asymptotic dimension explained

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. 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 l U of X such that every closed R -ball in X intersects at most n +1 subsets from l U . Here 'uniformly bounded' means that \sup _{U\in} \operatorname{diam}(U) < inf ty .

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): =inf ty otherwise.

Also, one says that a family

(X _{i)} _{i\in} 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 l U _{i} of X _{i} by sets of diameter at most D(R)< inf ty (independent of i ) such that every closed R -ball in X _{i} intersects at most n +1 subsets from l U _{i} .

Examples

• If

X is a metric space of bounded diameter then \operatorname{asdim}(X) =0 .\operatorname{asdim}( R)=\operatorname{asdim}( Z)=1 .\operatorname{asdim}( R ^{n)=n} .\operatorname{asdim}( 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 x 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} satisfies the inequality \operatorname{asdim} \le n uniformly. Then \operatorname{asdim}(X)\le \operatorname{asdim}(Y) + n. See^[2]

• If

X is a metric space with \operatorname{asdim}(X)< inf ty then X admits a coarse (uniform) embedding into a Hilbert space.^[3]

• 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.^[4]

Asymptotic dimension in geometric group theory

Asymptotic dimension achieved particular prominence in geometric group theory after a 1998 paper of Guoliang Yu^[5], 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)< inf ty , 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)< inf ty .^[8]

• If

G is relatively hyperbolic with respect to subgroups H _{1,...,} H _{k} each of which has finite asymptotic dimension then \operatorname{asdim}(G)< inf ty .^[9] \operatorname{asdim}( 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

asdim(F) =inf ty since F contains subgroups isomorphic to Z ^{n} for arbitrarily large n .

• If

G is the fundamental group of a finite graph of groups A with underlying graph A and finitely generated vertex groups, then^[10] $$\backslash operatorname(G)\backslash le\; 1+\; \backslash max\_\; \backslash operatorname\; (A\_v).$$

• Mapping class groups of orientable finite type surfaces have finite asymptotic dimension.^[11]
• Let

G be a connected Lie group and let \Gamma\le G be a finitely generated discrete subgroup. Then asdim(\Gamma)< inf ty .^[12]

• It is not known if Out(F _{n)} has finite asymptotic dimension for

n> 2 .^[13]

Further reading

• Gregory . Bell . Alexander . Dranishnikov . Asymptotic dimension . . 155 . 12 . 1265–96 . 2008 . 10.1016/j.topol.2008.02.011 . math/0703766.
• Book: Sergei . Buyalo . Viktor . Schroeder . Elements of Asymptotic Geometry . 2007 . European Mathematical Society . 978-3-03719-036-4 . EMS Monographs in Mathematics.

Notes and References

1. Book: Gromov, Mikhael . Asymptotic Invariants of Infinite Groups . Geometric Group Theory . 1993 . Cambridge University Press . 978-0-521-44680-8 . 2 . London Mathematical Society Lecture Note Series . 182.
2. 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 . 11 . 4749–64 . 2006 . 10.1090/S0002-9947-06-04088-8 . 2231870. free .
3. Book: Roe, John . Lectures on Coarse Geometry . 2003 . American Mathematical Society . 978-0-8218-3332-2 . University Lecture Series . 31.
4. Alexander . Dranishnikov . On hypersphericity of manifolds with finite asymptotic dimension . Transactions of the American Mathematical Society . 355 . 1 . 155–167 . 2003 . 10.1090/S0002-9947-02-03115-X . 1928082. free .
5. G. . Yu . 17189763 . The Novikov conjecture for groups with finite asymptotic dimension . Annals of Mathematics . 147 . 2 . 325–355 . 1998 . 121011. 10.2307/121011 .
6. Alexander . Dranishnikov . Асимптотическая топология . Asymptotic topology . Uspekhi Mat. Nauk . 55 . 6 . 71–16 . 2000 . 10.4213/rm334 . Russian. free .
   Alexander . Dranishnikov . Asymptotic topology . Russian Mathematical Surveys . 55 . 6 . 1085–1129 . 2000 . 10.1070/RM2000v055n06ABEH000334 . math/9907192. 2000RuMaS..55.1085D . 250889716 .
7. Guoliang . Yu . The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space . Inventiones Mathematicae . 139 . 1 . 201–240 . 2000 . 10.1007/s002229900032 . 2000InMat.139..201Y . 264199937 .
8. John . Roe . Hyperbolic groups have finite asymptotic dimension . Proceedings of the American Mathematical Society . 133 . 9 . 2489–90 . 2005 . 10.1090/S0002-9939-05-08138-4 . 2146189. free .
9. Densi . Osin . Asymptotic dimension of relatively hyperbolic groups . International Mathematics Research Notices . 2005 . 35 . 2143–61 . 2005 . 10.1155/IMRN.2005.2143 . math/0411585 . . 16743152 .
10. G. . Bell . A. . Dranishnikov . On asymptotic dimension of groups acting on trees . Geometriae Dedicata . 103 . 1 . 89–101 . 2004 . 10.1023/B:GEOM.0000013843.53884.77 . math/0111087. 14631642 .
11. Mladen . Bestvina . Koji . Fujiwara . Bounded cohomology of subgroups of mapping class groups . Geometry & Topology . 6 . 69–89 . 2002 . 1 . math/0012115. 10.2140/gt.2002.6.69 . 11350501 .
12. Lizhen . Ji . Asymptotic dimension and the integral K-theoretic Novikov conjecture for arithmetic groups . Journal of Differential Geometry . 68 . 3 . 535–544 . 2004 . 10.4310/jdg/1115669594. free .
13. Karen . Vogtmann . On the geometry of Outer space . Bulletin of the American Mathematical Society . 52 . 1 . 27–46 . 2015 . 10.1090/S0273-0979-2014-01466-1 . 3286480. free . Ch. 9.1