Combing nilpotent and polycyclic groups

Robert H. Gilman, Derek F. Holt and Sarah Rees


combings, automatic groups, nilpotent groups, polycyclic groups, formal languages, fellow traveller property


Published in Internat. J. Algebra Comput. 9 (1999) 135--155.


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

The preprint is available as gzipped dvi (39 kB) and postscript (128 kB) files.

Alternatively, you can request a copy by e-mailing me.

Sarah Rees
21 July 1998