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[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 ${\displaystyle X}$ be a metric space and ${\displaystyle n\geq 0}$ be an integer. We say that ${\displaystyle \operatorname {asdim} (X)\leq n}$ if for every ${\displaystyle R\geq 1}$ there exists a uniformly bounded cover ${\displaystyle {\mathcal {U}}}$ of ${\displaystyle X}$ such that every closed ${\displaystyle R}$-ball in ${\displaystyle X}$ intersects at most ${\displaystyle n+1}$ subsets from ${\displaystyle {\mathcal {U}}}$. Here 'uniformly bounded' means that ${\displaystyle \sup _{U\in {\mathcal {U}}}\operatorname {diam} (U)<\infty }$.

We then define the asymptotic dimension ${\displaystyle \operatorname {asdim} (X)}$ as the smallest integer ${\displaystyle n\geq 0}$ such that ${\displaystyle \operatorname {asdim} (X)\leq n}$, if at least one such ${\displaystyle n}$ exists, and define ${\displaystyle \operatorname {asdim} (X):=\infty }$ otherwise.

Also, one says that a family ${\displaystyle (X_{i})_{i\in I}}$ of metric spaces satisfies ${\displaystyle \operatorname {asdim} (X)\leq n}$ uniformly if for every ${\displaystyle R\geq 1}$ and every ${\displaystyle i\in I}$ there exists a cover ${\displaystyle {\mathcal {U}}_{i}}$ of ${\displaystyle X_{i}}$ by sets of diameter at most ${\displaystyle D(R)<\infty }$ (independent of ${\displaystyle i}$) such that every closed ${\displaystyle R}$-ball in ${\displaystyle X_{i}}$ intersects at most ${\displaystyle n+1}$ subsets from ${\displaystyle {\mathcal {U}}_{i}}$.

Examples

• If ${\displaystyle X}$ is a metric space of bounded diameter then ${\displaystyle \operatorname {asdim} (X)=0}$.
• ${\displaystyle \operatorname {asdim} (\mathbb {R} )=\operatorname {asdim} (\mathbb {Z} )=1}$.
• ${\displaystyle \operatorname {asdim} (\mathbb {R} ^{n})=n}$.
• ${\displaystyle \operatorname {asdim} (\mathbb {H} ^{n})=n}$.

Properties

• If ${\displaystyle Y\subseteq X}$ is a subspace of a metric space ${\displaystyle X}$, then ${\displaystyle \operatorname {asdim} (Y)\leq \operatorname {asdim} (X)}$.
• For any metric spaces ${\displaystyle X}$ and ${\displaystyle Y}$ one has ${\displaystyle \operatorname {asdim} (X\times Y)\leq \operatorname {asdim} (X)+\operatorname {asdim} (Y)}$.
• If ${\displaystyle A,B\subseteq X}$ then ${\displaystyle \operatorname {asdim} (A\cup B)\leq \max\{\operatorname {asdim} (A),\operatorname {asdim} (B)\}}$.
• If ${\displaystyle f:Y\to X}$ is a coarse embedding (e.g. a quasi-isometric embedding), then ${\displaystyle \operatorname {asdim} (Y)\leq \operatorname {asdim} (X)}$.
• If ${\displaystyle X}$ and ${\displaystyle Y}$ are coarsely equivalent metric spaces (e.g. quasi-isometric metric spaces), then ${\displaystyle \operatorname {asdim} (X)=\operatorname {asdim} (Y)}$.
• If ${\displaystyle X}$ is a real tree then ${\displaystyle \operatorname {asdim} (X)\leq 1}$.
• Let ${\displaystyle f:X\to Y}$ be a Lipschitz map from a geodesic metric space ${\displaystyle X}$ to a metric space ${\displaystyle Y}$ . Suppose that for every ${\displaystyle r>0}$ the set family ${\displaystyle \{f^{-1}(B_{r}(y))\}_{y\in Y}}$ satisfies the inequality ${\displaystyle \operatorname {asdim} \leq n}$ uniformly. Then ${\displaystyle \operatorname {asdim} (X)\leq \operatorname {asdim} (Y)+n.}$ See[3]
• If ${\displaystyle X}$ is a metric space with ${\displaystyle \operatorname {asdim} (X)<\infty }$ then ${\displaystyle X}$ admits a coarse (uniform) embedding into a Hilbert space.[4]
• If ${\displaystyle X}$ is a metric space of bounded geometry with ${\displaystyle \operatorname {asdim} (X)\leq n}$ then ${\displaystyle X}$ admits a coarse embedding into a product of ${\displaystyle n+1}$ 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 ${\displaystyle G}$ 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 ${\displaystyle \operatorname {asdim} (G)<\infty }$, then ${\displaystyle G}$ 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 ${\displaystyle G}$ is a word-hyperbolic group then ${\displaystyle \operatorname {asdim} (G)<\infty }$.[8]
• If ${\displaystyle G}$ is relatively hyperbolic with respect to subgroups ${\displaystyle H_{1},\dots ,H_{k}}$ each of which has finite asymptotic dimension then ${\displaystyle \operatorname {asdim} (G)<\infty }$.[9]
• ${\displaystyle \operatorname {asdim} (\mathbb {Z} ^{n})=n}$.
• If ${\displaystyle H\leq G}$, where ${\displaystyle H,G}$ are finitely generated, then ${\displaystyle \operatorname {asdim} (H)\leq \operatorname {asdim} (G)}$.
• For Thompson's group F we have ${\displaystyle asdim(F)=\infty }$ since ${\displaystyle F}$ contains subgroups isomorphic to ${\displaystyle \mathbb {Z} ^{n}}$ for arbitrarily large ${\displaystyle n}$.
• If ${\displaystyle G}$ is the fundamental group of a finite graph of groups ${\displaystyle \mathbb {A} }$ with underlying graph ${\displaystyle A}$ and finitely generated vertex groups, then[10]

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

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

References

1. 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.
2. 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.
3. 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.
4. Roe, John (2003). Lectures on Coarse Geometry. University Lecture Series. Vol. 31. American Mathematical Society. ISBN 978-0-8218-3332-2.
5. 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.
6. 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.
7. 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. CiteSeerX 10.1.1.155.1500. doi:10.1007/s002229900032. S2CID 264199937.
8. 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.
9. 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.
10. 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.
11. 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.
12. 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.
13. 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

Further reading

• Bell, Gregory; Dranishnikov, Alexander (2008). "Asymptotic dimension". Topology and Its Applications. 155 (12): 1265–96. arXiv:math/0703766. doi:10.1016/j.topol.2008.02.011.
• Buyalo, Sergei; Schroeder, Viktor (2007). Elements of Asymptotic Geometry. EMS Monographs in Mathematics. European Mathematical Society. ISBN 978-3-03719-036-4.