Preserving coarse properties
The aim of this paper is to investigate properties preserved and co-preserved by coarsely n-to-1 functions, in particular by the quotient maps \(X\rightarrow X/\sim \) induced by a finite group G acting by isometries on a metric space X. The coarse properties we are mainly interested in are related to asymptotic dimension and its generalizations: having finite asymptotic dimension, asymptotic Property C (as defined by Dranishnikov in Rus. Math. Surv. 55(6):1085–1129, 2000), straight finite decomposition complexity, countable asymptotic dimension, and metric sparsification property. We provide an alternative description of asymptotic Property C and we prove that the class of spaces with straight finite decomposition complexity coincides with the class of spaces of countable asymptotic dimension.
KeywordsAsymptotic dimension Asymptotic Property C Coarse geometry Coarsely n-to-1 functions Lipschitz maps Metric sparsification property Straight finite decomposition complexity
Mathematics Subject ClassificationPrimary 54F45 Secondary 55M10
- 6.Dranishnikov, A., Zarichnyi, M.: Asymptotic dimension, decomposition complexity, and Haver’s property C. arXiv:1301.3484
- 7.Dydak, J.: Coarse amenability and discreteness. arXiv:1307.3943
- 8.Dydak, J., Virk, Ž.: Inducing maps between Gromov boundaries. Mediterr J Math (accepted). arXiv:1506.08280
- 9.Engelking, R.: Theory of Dimensions Finite and Infinite, Sigma Series in Pure Mathematics, vol. 10. Heldermann Verlag, Berlin (1995)Google Scholar