Restrictions on Potential Automatic Structures on Thompson's Group F

TL;DR

This paper demonstrates that many languages close to geodesic representatives cannot form an automatic structure for Thompson's Group F, by showing length restrictions lead to contradictions in the Cayley graph.

Contribution

It introduces new restrictions on automatic structures for Thompson's Group F based on length and path analysis in the Cayley graph.

Findings

Certain languages cannot be part of an automatic structure for F.

Length restrictions lead to contradictions in path choices.

Automatic structures are limited by these geometric constraints.

Abstract

We show that a large class of languages in the standard finite generating set X = {$x_0, x_1, x_0^{-1}, x_1^{-1}$} cannot be part of an automatic structure for Thompson's Group F. These languages are ones that accept at least one representative of each element of F of word length that is within a fixed constant of a geodesic representative of the element. To accomplish this, we look at a specific element of F and trace two different paths through the Cayley graph to that element. We show that staying within the length restrictions along these two paths would force that element to have contradictory properties.