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
[edit]Let be a metric space and be an integer. We say that 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 .
We then define the asymptotic dimension as the smallest integer such that , if at least one such exists, and define otherwise.
Also, one says that a family of metric spaces satisfies 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
[edit]- If is a metric space of bounded diameter then .
- .
- .
- .
Properties
[edit]- If is a subspace of a metric space , then .
- For any metric spaces and one has .
- If then .
- If is a coarse embedding (e.g. a quasi-isometric embedding), then .
- If and are coarsely equivalent metric spaces (e.g. quasi-isometric metric spaces), then .
- If is a real tree then .
- Let 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 See[3]
- If is a metric space with then admits a coarse (uniform) embedding into a Hilbert space.[4]
- If is a metric space of bounded geometry with then admits a coarse embedding into a product of locally finite simplicial trees.[5]
Asymptotic dimension in geometric group theory
[edit]Asymptotic dimension achieved particular prominence in geometric group theory after a 1998 paper of Guoliang Yu[2] , 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 , 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.
- If is a word-hyperbolic group then .[8]
- If is relatively hyperbolic with respect to subgroups each of which has finite asymptotic dimension then .[9]
- .
- If , where are finitely generated, then .
- For Thompson's group F we have since contains subgroups isomorphic to for arbitrarily large .
- If is the fundamental group of a finite graph of groups with underlying graph and finitely generated vertex groups, then[10]
- Mapping class groups of orientable finite type surfaces have finite asymptotic dimension.[11]
- Let 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]
References
[edit]- ^ 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.
- ^ a b 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.
- ^ 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.
- ^ Roe, John (2003). Lectures on Coarse Geometry. University Lecture Series. Vol. 31. American Mathematical Society. ISBN 978-0-8218-3332-2.
- ^ 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.
- ^ 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. - ^ 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. doi:10.1007/s002229900032. S2CID 264199937.
- ^ 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.
- ^ 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.
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: CS1 maint: unflagged free DOI (link) - ^ 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.
- ^ 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.
- ^ 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.
- ^ 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
[edit]- 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.