Quasi - Isometrically Embedded Subgroups of Thompson ’ S Group F 3

The goal of this paper is to construct quasi-isometrically embedded subgroups of Thompson’s group F which are isomorphic to F × Zn for all n. A result estimating the norm of an element of Thompson’s group is found. As a corollary, Thompson’s group is seen to be an example of a finitely presented group which has an infinite-dimensional asymptotic cone. The interesting properties of Thompson’s group F have made it a favorite object of study among group theorists and topologists. It was first used by McKenzie and Thompson to construct finitely presented groups with unsolvable word problems ([5]). It is of interest also in homotopy theory in work related to homotopy idempotents, due to its universal conjugacy idempotent map φ, also used to see that F is an infinitely iterated HNN extension. In [1] Brown and Geoghegan found F to be the first torsion-free infinite-dimensional FP∞ group. Also, F contains an abelian free group of infinite rank, but it does not admit a free non-abelian subgroup. Many questions about F are still open, in particular it is not known whether F is automatic, or what is its Dehn function —although some estimates have been found by Guba, who proves it is polynomial in [4]—. The amenability of F is also unknown, fact that is of considerable interest since both the affirmative or negative answer would provide counterexamples to open questions (see [2]). Questions about the geometric properties of F have also been proposed. Bridson raised the question of whether F and F × Z could be quasi-isometric. In this paper we provide a partial answer to this question, proving that F admits a quasiisometrically embedded subgroup isomorphic to F × Z. As a corollary, F is the first example of a finitely presented group whose asymptotic cone is infinite-dimensional (see [3]). There are in the literature several interpretations of F that are useful to study it. Cannon, Floyd and Parry provide two of these interpretations, one as a group of homeomorphisms of the interval [0, 1], and another as a group of isomorphisms of rooted binary trees. In [1] Brown and Geoghegan prove that F is isomorphic to certain group of piecewise linear homeomorphisms of R, fact that will be used extensively in this paper. This construction allows us to translate group-theoretical questions to the setting of these homeomorphisms of R. 1