Floyd maps for relatively hyperbolic groups

Let ${\mathbf{delta}_{\mathcal S,\lambda}}$ denote the Floyd metric on a discrete group G generated by a finite set ${\mathcal S}$ with respect to the scaling function f _n ?= λ ⁿ for a positive λ <?1. We prove that if G is relatively hyperbolic with respect to a collection ${\mathcal P}$ of subgroups then there exists λ such that the identity map ${G\to G}$ extends to a continuous equivariant map from the completion with respect to ${\mathbf{\delta}_{\mathcal S,\lambda}}$ to the Bowditch completion of G with respect to ${\mathcal P}$ . In order to optimize the proof and the usage of the map theorem we propose two new definitions of relative hyperbolicity equivalent to the other known definitions. In our approach some “visibility” conditions in graphs are essential. We introduce a class of “visibility actions” that contains the class of relatively hyperbolic actions. The convergence property still holds for the visibility actions. Let a locally compact group G act on a compactum Λ with convergence property and on a locally compact Hausdorff space Ω properly and cocomactly. Then the topologies on Λ and Ω extend uniquely to a topology on the direct union ${T=\Lambda{\sqcup}\Omega}$ making T a compact Hausdorff space such that the action ${G{\curvearrowright}T}$ has convergence property. We call T the attractor sum of Λ and Ω.