Ian Chiswell

Embedding theorems for tree-free groups

We establish two embedding theorems for tree-free groups. The first result embeds a group G acting freely and without inversions on a � -tree X into a groupacting freely, without inversions, and transitively on a � -treein such a way that X embeds intoby means of a G-equivariant isometry. The second result embeds a group G acting freely and transitively on an R-tree X into RF(H ) for some suitable group H , again in such a way that X embeds G-equivariantly into the R-tree XH associated with RF(H ). The group RF(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 RF-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.