Thompson’s Group and the Linear Group GL∞（Z）

The authors study the finite decomposition complexity of metric spaces of H,equipped with different metrics,where is a subgroup of the linear group GL∞ （Z）.It is proved that there is an injective Lipschitz mapφ:（F,ds） → （H,d）,where is the Thompson’s group,the word-metric of with respect to the finite generating set and a metric of H.But it is not a proper map.Meanwhile,it is proved that:φ（F,ds） → （H,d1） is not a Lipschitz map,where d1 is another metric of H.