Embedding theorems for tree-free groups

Abstract

We establish two embedding theorems for tree-free groups. The first result embeds a group G acting freely and without inversions on a Λ-tree X into a group acting freely, without inversions, and transitively on a Λ-tree in such a way that X embeds into by means of a G-equivariant isometry. The second result embeds a group G acting freely and transitively on an ℝ-tree X into (H) for some suitable group H, again in such a way that X embeds G-equivariantly into the ℝ-tree XH associated with (H). The group (H) referred to here belongs to a class of groups introduced and studied by the present authors in [3]. As a consequence of these two theorems, we find that -groups and their associated ℝ-trees are in fact universal for free ℝ-tree actions. Moreover, our first embedding theorem throws light on the question, arising from the results of [7], whether a group endowed with a Lyndon length function L can always be embedded in a length-preserving way into a group with a regular Lyndon length function; modulo an obvious necessary restriction we show that this is indeed the case if L is free.

$Z^n$-free groups are CAT(0)

“…The structure theorem describes a finitely generated regular Z n -free group as a group obtained from a finitely generated free group by a sequence of finitely many HNN-extensions of a particular type. The same authors showed in [14] that every finitely generated Z n -free group embeds by a length-preserving monomorphism into a finitely generated regular Z m -free group, for some m. (Notice that by [5], every Λ-free group embeds by a length-preserving monomorphism into a regular Λ-free group.) We use these theorems to prove our main result (cf.…”

Actions, Length Functions, and Non-Archimedean Words

“…Let G be a finitely presented group with a free length function in Λ (not necessary regular). It can be embedded isometrically in the group G with a free regular length function in Λ by [27]. That group can be embedded in R(Λ ′ , X).…”

A hyperbolicity criterion for subgroups of $\mathcal{RF}(G)$

“…In a recent paper, Ian Chiswell and the present author show (among other things) that the groups RF (G) and their associated Rtrees X G , whose theory is developed in the forthcoming monographs [3,6], are universal with respect to inclusion for free R-tree actions; cf. [2,Theorem 8.6]. Since R = (R, +) acts on itself (viewed as an R-tree) freely by translation, and as the class of R-free groups is well known to be closed under taking free products (see [5,Prop.…”