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

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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

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  • If   is a metric space of bounded diameter then  .
  •  .
  •  .
  •  .

Properties

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  • 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

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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

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  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. ^ 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.
  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.{{cite journal}}: CS1 maint: unflagged free DOI (link)
  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

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