Cyclic Subgroups are Quasi-isometrically Embedded in the Thompson-Stein Groups

We give criteria for determining the approximate length of elements in any given cyclic subgroup of the Thompson-Stein groups F (n1, ..., nk) such that n1 − 1|ni − 1 ∀i ∈ {1, ..., k} in terms of the number of leaves in the minimal tree-pair diagram representative. This leads directly to the result that cyclic subgroups are quasi-isometrically embedded in the ThompsonStein groups. This result also leads to the corollaries that Zn is also quasiisometrically embedded in the Thompson-Stein groups for all n ∈ N and that the Thompson-Stein groups have infinite dimensional asymptotic cone.