Some Remarks on Subgroups of Hyperbolic Groups

If G is a hyperbolic group, where G = H oφ Z, H is finitely presented, and φ is an automorphism of H, then H satisfies a polynomial isoperimetric inequality. Necessary and sufficient conditions of homological character are given for a finitely presented subgroup H of a hyperbolic group to be hyperbolic (resp. a quasi-convex subgroup). If Y is a connected subcomplex of the finite connected 2complex X, where X/Y is of strictly negative curvature (in the sense of the weight test), then π1(Y ) is hyperbolic iff π1(X) is hyperbolic, and in this situation the pair (π1(X), π1(Y )) is relatively hyperbolic in the sense of Farb. A relation between these results and the Whitehead asphericity question is discussed. §