Thompson's group F and the linear group GL a (a"currency sign)

The authors study the finite decomposition complexity of metric spaces of H, equipped with different metrics, where H is a subgroup of the linear group GL(a)(a"currency sign). It is proved that there is an injective Lipschitz map phi: (F, d (S) ) -> (H, d), where F is the Thompson's group, dS the word-metric of F with respect to the finite generating set S and d a metric of H. But it is not a proper map. Meanwhile, it is proved that phi: (F, d (S) ) -> (H, d (1)) is not a Lipschitz map, where d (1) is another metric of H.