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 Thompson-Stein groups. This result also leads to the corollaries that ℤn is also quasi-isometrically embedded in the Thompson-Stein groups for all n ∈ ℕ and that the Thompson-Stein groups have infinite dimensional asymptotic cone. [ABSTRACT FROM AUTHOR]