Extending the synchronous fellow traveler property (Q6651098)

A normal form of a group satisfies the \textit{fellow traveler property} if every two normal forms of group elements which are at distance one (with respect to some fixed set of generators) are \(k\)-fellow travelers for some positive integer \(k\).\N\NIn the paper under review, the authors consider a relaxation of the fellow traveler property requiring fellow travelers to move with the same speed (synchronously) but allowing them to be at distance bounded from above by \(f(n)\), where \(f: \mathbb{N} \rightarrow \mathbb{R}_{+}\) is a function growing slower than any linear function and \(n\) is the distance that fellow travelers traversed starting from the origin (see Definition 3.1). They study normal forms satisfying this extended fellow traveler property and certain geometric constraints that naturally generalize two fundamental properties of an automatic normal form: the regularity of its language and the bounded length difference property.