Ending laminations for hyperbolic group extensions

TLDR: In this paper, it was shown that a hyperbolic automorphism cannot preserve any splitting over cyclic subgroups and that the limiting action is in fact free, whereas the collection of normal subgroups possible is limited, however, the class of groups G can still be fairly large.

Abstract: LetH be a hyperbolic normal subgroup of infinite index in a hyperbolic group G. It follows from work of Rips and Sela [16] (see below), that H has to be a free product of free groups and surface groups if it is torsion-free. From [14], the quotient group Q is hyperbolic and contains a free cyclic subgroup. This gives rise to a hyperbolic automorphism [2] of H . By iterating this automorphism, and scaling the Cayley graph of H , we get a sequence of actions of H on δi-hyperbolic metric spaces, where δi → 0 as i → ∞. From this, one can extract a subsequence converging to a small isometric action on a 0-hyperbolic metric space, i.e. an R-tree. By the JSJ splitting of Rips and Sela [16], [17], the outer automorphism group of H is generated by internal automorphisms. One notes further, that a hyperbolic automorphism cannot preserve any splitting over cyclic subgroups and that the limiting action is in fact free. Hence, by a theorem of Rips [16], H has to be a free product of free groups and surface groups if it is torsion-free. Thus the collection of normal subgroups possible is limited. However, the class of groups G can still be fairly large. Examples can be found in [3], [5] and [13]. For the purposes of this paper we choose a finite generating set of G that contains a finite generating set of H . Let ΓG and ΓH be the Cayley graphs of G, H with respect to these generating sets. There is a continuous proper embedding i of ΓH into ΓG. Every hyperbolic group admits a compactification of its Cayley graph by adjoining the Gromov boundary consisting of