Combings of groups and the grammar of reparameterization

Abstract

Abstract.A new construction of combings is used to distinguish between several previously indistinguishable classes of groups associated to the theory of automatic groups and non-positive curvature in group theory. We construct synchronously bounded combings for a class of groups that are neither bicombable nor automatic. The linguistic complexity of these combings is analysed: in many cases the language of words in the combing is an indexed language.

From automatic structures to automatic groups

“…In general combable groups are less amenable to computation than automatic groups. We refer to the work of Bridson [9] for an account of the relation between combable and automatic groups. On the other hand, a more relaxed fellow traveller property, called asynchronous fellow travelers, was introduced at the very beginning, see the book [15].…”

The conjugacy and isomorphism problems for combable groups

“…Each of these classes is defined in terms of a convexity condition, the common core of which is the "fellow-traveller" condition that forms the definition of a combable group: a group Γ with finite generating set A is said to be combable if there is a family of words {σ γ | γ ∈ Γ} in the letters A ±1 and a constant k > 0 such that for each γ ∈ Γ and a ∈ A, the paths in the Cayley graph C A Γ that begin at the identity vertex and are labelled σ γ and σ γa remain uniformly k-close. The results in [5] established that the class of combable groups is strictly larger than all of the other classes listed above. There is an effective solution to the word problem in any combable group [8], [4].…”

“…, but there is no algorithm to decide which are isomorphic to Γ 1 . In order to obtain such sequences we combine the construction of [5] with a suitable encoding of the fact that there is no algorithm to decide which m-element subsets generate a direct product of free groups, and likewise for certain hyperbolic groups.…”

Banach strong Novikov conjecture for polynomially contractible groups

“…Then we may apply this string of generators to any other element of the group. There exist groups which are combable but not bi-combable, see Bridson [Bri03].…”