REGULAR COCYCLES AND BIAUTOMATIC STRUCTURES

Abstract

In [ECHLPT] and [S] it is shown that if the fundamental group of a Seifert fibred 3manifold is not virtually nilpotent then it has an automatic structure. In the unpublished 1992 preprint [G2] Gersten constructs a biautomatic structure on the fundamental group of any circle bundle over a hyperbolic surface. He asks if the same can be done for the above Seifert fibered 3-manifold. We show the existence of such a biautomatic structure. We do this in the context of a general discussion of biautomatic structures on virtually central extensions of finitely generated groups. A virtually central extension is an extension of a group by an abelian group for which the induced action of on is finite, that is, given by a map with finite image. The fundamental group of a Seifert fibered 3-manifold as above is a virtually central extension of a Fuchsian group by . (For convenience we are using the term “Fuchsian group” for any discrete finitely generated subgroup of — orientable or not.) We use a concept of “regular 2-cocycles” on a group which was suggested by Gersten’s work. Here “regularity” is with respect to a (poss ibly asynchronously) automatic structure on . If is a biautomatic structure on we show that any virtually central extension of defined by an -regular cocycle also has a biautomatic structure. As an application we show that any virtually central extension of a Fuchsian group by a finitely generated abelian group is biautomatic. In fact, if is a geodesic language on , we show that all of is represented by -regular cocycles . In case is torsion free and with trivial -action this is implicit in Gersten’s work (loc. cit. — we give an independent treatment here that is mor e geometric; alternatively, it follows from his result about biautomaticity plus Theorem A below). The general case follows easily from this using Corollary 2.7 below, which says that a cohomology class for a group is regular if its restriction to some finite index subgroup of is regular. The converse to the fact that regular cocycles lead to biautomatic structures is also true. Theorem A. Let be a virtually central extension of the group by a finitely generated abelian group . Then carries a biautomatic structure if and only if has a biautomatic structure for which the cohomology class of the extension is represent ed by an -regular cocycle.