Groups with Finitely Many Shortlex Cones

We show that a finitely generated group $G=⟨\mathrm{\Sigma }⟩$ satisfying the falsification by fellow traveller property also satisfies the shortlex falsification by fellow traveller property. This implies that there are finitely many shortlex cone types and, therefore, that the corresponding language of shortlex representatives ${\text{SL}}_G\subset (\mathrm{\Sigma }\cup {\mathrm{\Sigma }}^{-1}{)}^{\ast }$ is a regular language. Following the example of M. Elder, we prove that the converse of the previous statements do not hold.