Combing Nilpotent and Polycyclic Groups

The notable exclusions from the family of automatic groups are those nilpotent groups which are not virtually abelian, and the fundamental groups of compact 3-manifolds based on the Nil or Sol geometries. Of these, the 3-manifold groups have been shown by Bridson and Gilman to lie in a family of groups defined by conditions slightly more general than those of automatic groups, that is, to have combings which lie in the formal language class of indexed languages. In fact, the combings constructed by Bridson and Gilman for these groups can also be seen to be real-time languages (that is, recognised by real-time Turing machines). This article investigates the situation for nilpotent and polycyclic groups. It is shown that a finitely generated class 2 nilpotent group with cyclic commutator subgroup is real-time combable, as are also all 2 or 3-generated class 2 nilpotent groups, and groups in specific families of nilpotent groups (the finitely generated Heisenberg groups, groups of unipotent matrices over Z and the free class 2 nilpotent groups). Further it is shown that any polycyclic-by-finite group embeds in a real-time combable group. All the combings constructed in the article are boundedly asynchronous, and those for nilpotent-by-finite groups have polynomially bounded length functions. AMS subject classifications: 20F10, 20-04, 68Q40, secondary: 03D40 The first author was partially supported by NFS grant DMS-9401090. The second and third authors would like to thank the Fakultät für Mathematik of the Universität Bielefeld for its hospitality while this work was carried out, and the Deutscher Akademischer Austauschdienst and EPSRC for financial support.