Extending the Synchronous Fellow Traveler Property

We introduce an extension of the fellow traveler property which allows fellow travelers to be at distance bounded from above by a function $f(n)$ growing slower than any linear function. We 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. We show examples of such normal forms and prove some non-existence theorems.