On finite decomposition complexity of Thompson group

Finite decomposition complexity (FDC) is a large scale property of a metric space. It generalizes finite asymptotic dimension and applies to a wide class of groups. To make the property quantitative, a countable ordinal “the complexity” can be defined for a metric space with FDC. In this paper we prove that the subgroup Z≀Z of Thompsonʼs group F belongs to Dω exactly, where ω is the smallest infinite ordinal number and show that F equipped with the word-metric with respect to the infinite generating set {x0,x1,…,xn,…} does not have finite decomposition complexity.