Bounded Cohomology and Combings of Groups

We adopt the notion of combability of groups defined in [Ghys2]. An example is given of a combable group which is not residually finite. Two of the eight 3-dimensional geometries, e Sl2(R) and H ×R, are quasiisometric. Three dimensional geometries are classified up to quasiisometry. Seifert fibred manifolds over hyperbolic orbifolds have bicombable fundamental groups. Every combable group satisfies an exponential isoperimetric inequality. Central extension of Z by hyperbolic surface groups are biautomatic. Central extensions of Z by finitely presented C(1/4)–T(4) groups are biautomatic. Examples of group cohomology classes are produced all of whose cocycle representatives grow exponentially or superexponentially. Higher dimensional integral Heisenberg groups satisfy a quadratic weak isoperimetric inequality. A cohomology theory based on bounded valued cocycles is introduced for groups satisfying finiteness conditions to detect isoperimetric inequalities.