Central extensions of word hyperbolic groups

TLDR: In this paper, it was shown that a 2-dimensional cohomology class on a word hyperbolic group can be represented by a bounded 2-cocycle, which is a special case of the notion of weak boundedness.

Abstract: perbolic 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 > 2 is bounded. Our results apply more generally to virtually central extensions. We build on the ideas presented in [4], 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 [1], [2], and [7]. The new ingredient is a maximising technique inspired by work of Epstein and Fujiwara. Beginning with an arbitrary finite generating set for a central extension by 2, this maximising process is used to obtain a section which, in the language of [4], corresponds to a "regular 2-cocycle" on the hyperbolic group G, and can be used to obtain a biautomatic 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.