Metric spaces with complexity of the smallest infinite ordinal number (Q1694803)

This study is motivated by the property of finite asymptotic dimension due to M. Gromov. A (geometric) concept of finite decomposition complexity was introduced by \textit{E. Guentner} et al. [Groups Geom. Dyn. 7, No. 2, 377--402 (2013; Zbl 1272.52041)]. ``Finite decomposition complexity is a large scale property of a metric space. Roughly speaking, a metric space has finite decomposition complexity when there is an algorithm to decompose the space into nice pieces in a certain asymptotic way''. In this article, the authors are concerned with a study of the exact complexity of \(G_n\), and this is particularly motivated by the question of the finite decomposition complexity of Thomson's group \(F\). The authors prove in this article that the exact complexity of the finite product \(\mathbb{Z}\wr\mathbb{Z}\times\mathbb{Z}\wr\mathbb{Z}\times\cdots\times \mathbb{Z}\wr\mathbb{Z}\) of wreath products is \(\omega\), where \(\omega\) is the smallest infinite ordinal number (see Theorem 3.5) and deduce that the complexity of \((\mathbb{Z}\wr\mathbb{Z})\wr\mathbb{Z}\) is \(\omega+1\).