Distortion and `1-homology

If H is a nitely generated subgroup of the nitely generated group G, then H is undistorted in G for their word metrics ii the induced homomorphism on the zero-dimensionaì 1-homology group is injective. The Mayer-Vietoris exact sequence for`1-homology is established and applications are given to hyperbolic groups and relative hyperbolicity. The notion of relative distortion of two maps is introduced and a homological criterion for it is established. A nitely generated subgroup H of the nitely generated group G is said to be undistorted for their respective word metrics (a.k.a. the inclusion H < G is a quasi-isometric imbedding) if the distance between two elements of H, as measured in the word metric of G, is bilipschitz equivalent to the distance as measured in H. In Ge3] it was established that a nitely generated subgroup H of a hyperbolic group G is quasi-isometrically imbedded in G ii the restriction map on`1-cohomology H 1 (1) (H; Z) is surjective. Since the cohomological characterization of hyperbolic groups proved in Ge2], that a nitely presented group G is hyperbolic ii H 2 (1) (G; ` 1) = 0, was converted into the homological criterion in AG], that G is hyperbolic ii H (1) 2 (G; R) = H (1) 1 (G; R) = 0, it was natural to ask whether there was a corresponding homological characterization for a general nitely generated subgroup H of a nitely generated group G to be undistorted for their word metrics. We shall prove in Corollary 3.3 that if H is a nitely generated subgroup of the nitely generated group G, then the inclusion H < G is undistorted for their word metrics ii the induced map H (1) 0 (H; R) ! H (1) 0 (G; R) is injective. This result follows from a more general result Theorem 3.2 on graphs: If is a connected subgraph of the graph ?, then the inclusion ? does not distort their respective word metrics (i.e. the inclusion is a quasi-isometric imbedding) ii the induced map H (1) 0 ((; R) ! H (1) 0 (?; R) is injective. In x4 we prove a generalization of Theorem 3.2 that handles the case when is not connected. Of particular interest here is Theorem 4.7, which says, in the terminology of Deenition 4.8, that to detect distortion one need only look at unramiied boundaries. In x5 we establish a Mayer-Vietoris exact sequence …