Boundary Homeomorphisms and Quasi-isometries Between Hyperbolic Groups with Two-Ended Splittings

Christopher Cashen (University of Vienna)

We construct invariants for boundary homeomorphism and quasi-isometry of hyperbolic groups that split over two-ended subgroups in terms of the respective homeomorphism/quasi-isometry types of the vertex groups relative to their incident edge groups. For boundary homeomorphism we get a complete invariant. For quasi-isometry we get a complete invariant when the vertex groups are all either relatively rigid or hanging. Joint w/ Alexandre Martin