I. Mineyev, A. Yaman

Let G be a finitely generated group and G'={ G_i | i in I } be a family of its subgroups. We utilize the notion of a tuple T=(G,G',X,V') that makes the statements and arguments for the pair (G,G') parallel to the non-relative case. For a given tuple, we define snakes and the snake metric. The language of tuples and snakes seems to be convenient for dealing with relative hyperbolicity. For a tuple T we consider properties of being finitely generated, finitely presented, and of having fine triangles. The pair (G,G') is called hyperbolic if there is a finitely generated tuple (G,G',X,V') which has fine triangles and with X fine. We give a definition of relative hyperbolicity which slightly generalizes the definition of Bowditch, and show that (G,G') is hyperbolic iff G is relatively hyperbolic with respect to G'. We generalize the argument given by Mineyev for hyperbolic groups to show that if (G, G') is hyperbolic then $H_b^{2}(G,G';V) \to H^{2}(G,G';V)$ is surjective for all bounded QG-modules V. The same is true for bounded RG-modules, bounded CG-modules, and Banach modules. Moreover, this statement extends to a characterization of hyperbolicity of (G,G').