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 the automatic groups, i.e. 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 (i.e. recognized 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 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, of a degree equal to the nilpotency class, c; this verifies a polynomial upper bound on the Dehn functions of those groups of degree c+1.