Central Extensions of Word Hyperbolic Groups

Thurston has claimed (unpublished) that central extensions of word hyperbolic groups by finitely generated abelian groups are automatic. We show that they are in fact biautomatic. Further, we show that every 2-dimensional cohomology class on a word hyperbolic group can be represented by a bounded 2-cocycle. This lends weight to the claim of Gromov that for a word hyperbolic group, the cohomology in every dimension is bounded. Our results apply more generally to virtually central extensions. We build on the ideas presented in [NR], where the general problem was reduced to the case of central extensions by Z and was solved for Fuchsian groups. Some special cases of automaticity or biautomaticity in this case had previously been proved in [ECHLPT], [Sha], and [Ge]. The new ingredient is a maximising technique inspired by work of Epstein and Fuji-wara. Beginning with an arbitrary finite generating set for a central extension by Z, this maximising process is used to obtain a section which, in the language of [NR], corresponds to a " regular 2-cocycle " on the hyperbolic group G, and can be used to obtain a biauto-matic structure for the extension. Since central extensions correspond to 2-dimensional cohomology classes, it follows that every such class can be represented by a regular 2-cocycle. Using the geometric properties of G, we then further modify this cocycle to obtain a bounded representative for the original cohomology class. We also discuss the relations between various concepts of " weak boundedness " of a 2-cocycle on an arbitrary finitely generated group, related to quasi-isometry properties of central extensions. For cohomology classes these weak boundedness concepts are shown to be equivalent to each other. We do not know if a weakly bounded cohomology class must be bounded. Let G be a finitely generated group and X a finite set which maps to a monoid generating set of G. The map of X to G can be extended in the obvious way to give a monoid homomorphism of X * onto G which will be denoted by w → w. For convenience of exposition we will always assume our generating sets are symmetric, that is, they satisfy X = X −1. If L ⊂ X * then the pair consisting of L and the evaluation map L → G will be called a language on G. Abusing terminology, we will often suppress the evaluation map and just call L …