1994/11/07 by Neumann, Walter D., Reeves, Lawrence
#FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.math/9411203
Let E be a virtually central extension of the group G by a finitely generated abelian group A. We show that E carries a biautomatic structure if and only if G has a biautomatic structure L for which the cohomology class of the extension is represented by an L-regular cocycle. Moreover, a cohomology class is L-regular if some multiple of it is or if its restriction to some finite index subgroup is. We also show that the entire second cohomology of a Fuchsian group is regular, so any virtually central extension is biautomatic. In particular, if the fundamental group of a Seifert fibered 3-manifold is not virtually nilpotent then it is biautomatic. ECHLPT had shown automaticity in this case and in an unpublished 1992 preprint Gersten constructed a biautomatic structure for circle bundles over hyperbolic surfaces and asked if the same could be done for these Seifert fibered 3-manifolds.