Regular Cocycles and Biautomatic Structures

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 In the unpublished 1992 preprint [G2] Gersten constructs a biautomatic structure on the fundamental group of any circle bundle over a hyperbolic surface.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 G by an abelian group A for which the induced action of G on A is finite, that is, given by a map G ! Aut(A) with finite image. The fundamental group of a Seifert fibered 3-manifold as above is a virtually central extension of a Fuchsian group G by Z. (For convenience we are using the term “Fuchsian group” for any discrete finitely generated subgroup of Isom(H2) — orientable or not.) We use a concept of “regular 2-cocycles” on a group G which was suggested by Gersten’s work. Here “regularity” is with respect to a (possibly asynchronously) automatic structure L on G. If L is a biautomatic structure on G we show that any virtuallycentral extension of G defined by an L-regular cocycle also has a biautomatic structure. As an application we show that any virtually central extension of a Fuchsian group G by a finitely generated abelian group A is biautomatic. In fact, if L is a geodesic language on G, we show that all of H2(G;A) is represented by L-regular cocycles1). In case G is torsion free and A =Zwith trivialG-action this is implicit in Gersten’s work (loc. cit. — we give an independent treatment here that is more 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 G is regular if its restriction to some finite index subgroup of G is regular. The converse to the fact that regular cocycles lead to biautomatic structures is also true. Theorem A. Let E be a virtually central extension of the group G by a finitely generated abelian groupA. ThenE carries a biautomatic structure if and only if Ghas a biautomatic structureL for which the cohomology class of the extension is represented by anL-regular cocycle.