We furnish an example of a finite generating set for a group that does not enjoy the falsification by fellow traveler property, while the full language of geodesics is regular.
"Regular geodesic languages and the falsification by fellow traveler property." Algebr. Geom. Topol. 5 (1) 129 - 134, 2005. https://doi.org/10.2140/agt.2005.5.129