A Remark about Combings of Groups

In the last several years a remarkable interplay between geometry, group theory, and the theory of formal languages has led to developments including the introduction of automatic groups [Ep+], hyperbolic groups [Gro], and geometric and language-theoretic characterizations of finitely generated virtually free groups [MS1] [MS2]. A prime example of such interplay is the study of combings, which were introduced in [Ep+, Chapter 3.6]. A combing is simply a choice of normal form for group elements relative to a fixed finite set of generators, but typically the set of words in normal form is required to satisfy language theoretic constraints when viewed as a formal language over the alphabet of generators, and geometric constraints when interpreted as a collection of paths in the Cayley graph of the group. For example, a group G is said to be automatic [Ep+] if for some finite set of generators it has a set of normal forms which is a regular language and has the property that the paths determined by normal forms for nearby elements are uniformly close, i.e., there exists a constant K > 0 such that points travelling at unit speed along two such paths which end a distance 1 apart stay K-close; this last condition is called the synchronous fellow traveller property. If these two conditions hold for one finite set of generators then they hold for all. Language theoretic and geometric conditions afford new ways of looking at finitely generated groups, and it is of interest to consider how various classes of groups may be characterized in terms of the combings which they admit. A well known open question in the study of automatic groups is whether or not one needs to include the language theoretic constraint of regularity in the definition. Might it be the case that every group which admits a set of normal forms satisfying the synchronous fellow traveller property is automatic, i.e., admits a regular set of normal forms satisfying the synchronous fellow traveller property? It seems highly unlikely that this will turn out to be the case, and various possible counterexamples have been proposed, including the three dimensional Heisenberg groupH3, and irreducible uniform lattices in Sl2(R)× Sl2(R). However, despite much effort, this question remains unresolved. What is known is that H3 provides a counterexample to the analogous question involving the asynchronous fellow traveller property. This