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\ge0

be an integer. We say that

\operatorname{asdim}(X)\len

if for every

R\ge1

there exists a uniformly bounded cover

lU

of

X

such that every closed

R

-ball in

X

intersects at most

n+1

subsets from

lU

. Here 'uniformly bounded' means that

\supU\in\operatorname{diam}(U)<infty

.

We then define the asymptotic dimension

\operatorname{asdim}(X)

as the smallest integer

n\ge0

such that

\operatorname{asdim}(X)\len

, if at least one such

n

exists, and define

\operatorname{asdim}(X):=infty

otherwise.

Also, one says that a family

(Xi)i\in

of metric spaces satisfies

\operatorname{asdim}(X)\len

uniformly if for every

R\ge1

and every

i\inI

there exists a cover

lUi

of

Xi

by sets of diameter at most

D(R)<infty

(independent of

i

) such that every closed

R

-ball in

Xi

intersects at most

n+1

subsets from

lUi

.

Examples

X

is a metric space of bounded diameter then

\operatorname{asdim}(X)=0

.

\operatorname{asdim}(R)=\operatorname{asdim}(Z)=1

.

\operatorname{asdim}(Rn)=n

.

\operatorname{asdim}(Hn)=n

.

Properties

Y\subseteqX

is a subspace of a metric space

X

, then

\operatorname{asdim}(Y)\le\operatorname{asdim}(X)

.

X

and

Y

one has

\operatorname{asdim}(X x Y)\le\operatorname{asdim}(X)+\operatorname{asdim}(Y)

.

A,B\subseteqX

then

\operatorname{asdim}(A\cupB)\lemax\{\operatorname{asdim}(A),\operatorname{asdim}(B)\}

.

f:Y\toX

is a coarse embedding (e.g. a quasi-isometric embedding), then

\operatorname{asdim}(Y)\le\operatorname{asdim}(X)

.

X

and

Y

are coarsely equivalent metric spaces (e.g. quasi-isometric metric spaces), then

\operatorname{asdim}(X)=\operatorname{asdim}(Y)

.

X

is a real tree then

\operatorname{asdim}(X)\le1

.

f:X\toY

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(Br(y))\}y\in

satisfies the inequality

\operatorname{asdim}\len

uniformly. Then

\operatorname{asdim}(X)\le\operatorname{asdim}(Y)+n.

See[2]

X

is a metric space with

\operatorname{asdim}(X)<infty

then

X

admits a coarse (uniform) embedding into a Hilbert space.[3]

X

is a metric space of bounded geometry with

\operatorname{asdim}(X)\len

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)<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.

G

is a word-hyperbolic group then

\operatorname{asdim}(G)<infty

.[8]

G

is relatively hyperbolic with respect to subgroups

H1,...,Hk

each of which has finite asymptotic dimension then

\operatorname{asdim}(G)<infty

.[9]

\operatorname{asdim}(Zn)=n

.

H\leG

, where

H,G

are finitely generated, then

\operatorname{asdim}(H)\le\operatorname{asdim}(G)

.

asdim(F)=infty

since

F

contains subgroups isomorphic to

Zn

for arbitrarily large

n

.

G

is the fundamental group of a finite graph of groups

A

with underlying graph

A

and finitely generated vertex groups, then[10] \operatorname(G)\le 1+ \max_ \operatorname (A_v).

G

be a connected Lie group and let

\Gamma\leG

be a finitely generated discrete subgroup. Then

asdim(\Gamma)<infty

.[12]

n>2

.[13]

Further reading

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