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 Thompson’s 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. © 2011 Elsevier Inc. All rights reserved.