Bounded group actions on trees and hyperbolic and Lyndon length functions (Q583364)

Let the group G act as a group of isometries on a metric tree T and let K be a normal subgroup with bounded action (that is the distance of any point to any of its K-images is bounded). The action of G gives a hyperbolic length function on G. It is shown that the hyperbolic length function on G is determined by that on G/K and that the Lyndon length functions on G are characterized up to equivalences by those on G/K. Moreover, normal subgroups with bounded action are associated to a length function and there is a maximal normal subgroup with bounded action. Finally, if not every element of G has a fixed point then, for K maximal, G/K is isomorphic to a subgroup of the additive reals or has trivial centre.