Matrix algebra of sets and variants of decomposition complexity. (English) Zbl 1443.54013
In this paper, the author introduces operations for indexed families of subsets with two variables, and proves some properties analogous to matrix algebra. By means of the operations, the author gives a characterization of asymptotic dimension of M. Gromov [Geometric group theory. Volume 2: Asymptotic invariants of infinite groups. Proceedings of the symposium held at the Sussex University, Brighton, July 14-19, 1991. Cambridge: Cambridge University Press (1993; Zbl 0841.20039)].
A new variant of decomposition complexity, called asymptotic property D, is also introduced. It is proved that for every \(\infty\)-pseudo-metric space (that is, a pseudo-metric space whose pseudo-metric admits the value \(\infty\)), finite asymptotic dimension implies asymptotic property D, and asymptotic property D implies asymptotic property C introduced by A. N. Dranishnikov [Russ. Math. Surv. 55, No. 6, 1085–1129 (2000; Zbl 1028.54032); translation from Usp. Mat. Nauk 55, No. 6, 71–116 (2000)].
As an application of the above notions, the author proves that countable reduced (restricted) direct products of uniformly discrete \(\infty\)-pseudo-metric spaces with finite asymptotic dimension have asymptotic property D. This improves a theorem of T. Davila [“On asymptotic property C”, Preprint, arXiv:1611.05988] stating that countable reduced (restricted) direct products of countable groups with finite asymptotic dimension have asymptotic property C.
A new variant of decomposition complexity, called asymptotic property D, is also introduced. It is proved that for every \(\infty\)-pseudo-metric space (that is, a pseudo-metric space whose pseudo-metric admits the value \(\infty\)), finite asymptotic dimension implies asymptotic property D, and asymptotic property D implies asymptotic property C introduced by A. N. Dranishnikov [Russ. Math. Surv. 55, No. 6, 1085–1129 (2000; Zbl 1028.54032); translation from Usp. Mat. Nauk 55, No. 6, 71–116 (2000)].
As an application of the above notions, the author proves that countable reduced (restricted) direct products of uniformly discrete \(\infty\)-pseudo-metric spaces with finite asymptotic dimension have asymptotic property D. This improves a theorem of T. Davila [“On asymptotic property C”, Preprint, arXiv:1611.05988] stating that countable reduced (restricted) direct products of countable groups with finite asymptotic dimension have asymptotic property C.
Reviewer: Takamitsu Yamauchi (Matsuyama)
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