Abstract
In this chapter we review the construction of the Higson compactification (and corona) and the concept of asymptotic dimension for metric spaces.
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Notes
- 1.
We only consider Hausdorff compactifications.
- 2.
All compactifications of X are ≤ X β, where X β is the Stone-Čech compactification. Thus we can assume that they are quotients of X β, and therefore they form a set.
- 3.
Recall that a function f : X →C vanishes at infinity when, for all ε > 0, there is a compact K ⊂ X so that |f(x)| < ε for all \(x\in X\smallsetminus K\).
- 4.
The original notation of Gromov [56] is \(\operatorname {as\,dim}_+M\).
References
J.A. Álvarez López, A. Candel, Algebraic characterization of quasi-isometric spaces via the Higson compactification. Topology Appl. 158(13), 1679–1694 (2011)
G. Bell, A. Dranishnikov, Asymptotic dimension in Bȩdlewo. Topology Proc. 38, 209–236 (2011)
A. Dranishnikov, J. Keesling, V. Uspenskij, On the Higson corona of uniformly contractible spaces. Topology 37(4), 791–803 (1998)
M. Gromov, Asymptotic invariants of infinite groups, in Geometric Group Theory, Vol. 2 (Sussex, 1991). London Mathematical Society Lecture Note Series, vol. 182 (Cambridge University Press, Cambridge, 1993), pp. 1–295
J. Roe, Lectures on Coarse Geometry. University Lecture Series, vol. 31 (American Mathematical Society, Providence, 2003)
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Álvarez López, J.A., Candel, A. (2018). Higson Corona and Asymptotic Dimension. In: Generic Coarse Geometry of Leaves. Lecture Notes in Mathematics, vol 2223. Springer, Cham. https://doi.org/10.1007/978-3-319-94132-5_7
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DOI: https://doi.org/10.1007/978-3-319-94132-5_7
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