Subgroups of Word Hyperbolic Groups in Dimension 2

Abstract

If G is a word hyperbolic group of cohomological dimension 2, then every subgroup of G of type FP2 is also word hyperbolic. Isoperimetric inequalities are defined for groups of type FP2 and it is shown that the linear isoperimetric inequality in this generalized context is equivalent to word hyperbolicity. A sufficient condition for hyperbolicity of a general graph is given along with an application to ‘relative hyperbolicity’. Finitely presented subgroups of Lyndon's small cancellation groups of hyperbolic type are word hyperbolic. Finitely presented subgroups of hyperbolic 1-relator groups are hyperbolic. Finitely presented subgroups of free Burnside groups are finite in the stable range.