On Transfinite Extension of Asymptotic Dimension

We prove that a transfinite extension of asymptotic dimension asind is trivial. We introduce a transfinite extension of asymptotic dimension asdim and give an example of metric proper space which has transfinite infinite dimension. 0. Asymptotic dimension asdim of a metric space was defined by Gromov for studying asymptotic invariants of discrete groups [1]. This dimension can be considered as asymptotic analogue of the Lebesgue covering dimension dim. Dran-ishnikov has introduced dimensions asInd and asind which are analogous to large inductive dimension Ind and small inductive dimension ind [2,3].It is known that asdim X = asInd X for each proper metric space with asdim X < ∞. The problem of coincidence of asdim and asInd is still open in the general case [3]. Extending codomain of Ind to ordinal numbers we obtain the transfinite extension trInd of the dimension Ind. It is known that there exists a space S α such that trInd S α = α for each countable ordinal number α [4]. Zarichnyi has proposed to consider transfinite extension of asInd and conjectured that this extension is trivial. It is proved in [5] that if a space has a transfinite asymptotic dimension trasInd, then this dimension is finite. We investigate in this paper transfinite extensions for the asymptotic dimensions asind and asdim. It appears that extending codomain of asind to ordinal numbers we obtain the trivial extension as well. However, the main result of this paper is construction of transfinite extension trasdim of asdim which is not trivial. Moreover , trasdim classifies the metric spaces with asymptotic property C introduced by Dranishnikov [8]. The paper is organized as follows: in Section 1 we give some necessary definitions and introduce some denotations, in Section 2 we prove that the transfinite extension