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
be a
metric space and
be an integer. We say that
\operatorname{asdim}(X)\len
if for every
there exists a uniformly bounded cover
of
such that every closed
-ball in
intersects at most
subsets from
. Here 'uniformly bounded' means that
\supU\in\operatorname{diam}(U)<infty
.
We then define the asymptotic dimension
as the smallest integer
such that
\operatorname{asdim}(X)\len
, if at least one such
exists, and define
\operatorname{asdim}(X):=infty
otherwise.
Also, one says that a family
of metric spaces satisfies
\operatorname{asdim}(X)\len
uniformly if for every
and every
there exists a cover
of
by sets of diameter at most
(independent of
) such that every closed
-ball in
intersects at most
subsets from
.
Examples
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
is a subspace of a metric space
, then
\operatorname{asdim}(Y)\le\operatorname{asdim}(X)
.
and
one has
\operatorname{asdim}(X x Y)\le\operatorname{asdim}(X)+\operatorname{asdim}(Y)
.
then
\operatorname{asdim}(A\cupB)\lemax\{\operatorname{asdim}(A),\operatorname{asdim}(B)\}
.
is a coarse embedding (e.g. a quasi-isometric embedding), then
\operatorname{asdim}(Y)\le\operatorname{asdim}(X)
.
and
are coarsely equivalent metric spaces (e.g. quasi-isometric metric spaces), then
\operatorname{asdim}(X)=\operatorname{asdim}(Y)
.
is a
real tree then
\operatorname{asdim}(X)\le1
.
be a Lipschitz map from a geodesic metric space
to a metric space
. Suppose that for every
the set family
satisfies the inequality
uniformly. Then
\operatorname{asdim}(X)\le\operatorname{asdim}(Y)+n.
See
[2]
is a metric space with
\operatorname{asdim}(X)<infty
then
admits a coarse (uniform) embedding into a Hilbert space.
[3]
is a metric space of bounded geometry with
\operatorname{asdim}(X)\len
then
admits a coarse embedding into a product of
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
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
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.
is a
word-hyperbolic group then
\operatorname{asdim}(G)<infty
.
[8]
is
relatively hyperbolic with respect to subgroups
each of which has finite asymptotic dimension then
\operatorname{asdim}(G)<infty
.
[9] \operatorname{asdim}(Zn)=n
.
, where
are finitely generated, then
\operatorname{asdim}(H)\le\operatorname{asdim}(G)
.
since
contains subgroups isomorphic to
for arbitrarily large
.
is the fundamental group of a finite
graph of groups
with underlying graph
and finitely generated vertex groups, then
[10]
be a connected
Lie group and let
be a finitely generated discrete subgroup. Then
.
[12] - It is not known if
has finite asymptotic dimension for
.
[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
- 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.
- 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 .
- Book: Roe, John . Lectures on Coarse Geometry . 2003 . American Mathematical Society . 978-0-8218-3332-2 . University Lecture Series . 31.
- 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 .
- G. . Yu . 17189763 . The Novikov conjecture for groups with finite asymptotic dimension . Annals of Mathematics . 147 . 2 . 325–355 . 1998 . 121011. 10.2307/121011 .
- 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 .
- 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 .
- 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 .
- 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 .
- 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 .
- 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 .
- 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 .
- 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