  In this article we show that every group with a finite presentation
satisfying one or both of the small cancellation conditions $C'(1/6)$ and
$C'(1/4)-T(4)$ has the property that the set of all geodesics (over the same
generating set) is a star-free regular language. Star-free regularity of the
geodesic set is shown to be dependent on the generating set chosen, even for
free groups. We also show that the class of groups whose geodesic sets are
star-free with respect to some generating set is closed under taking graph (and
hence free and direct) products, and includes all virtually abelian groups.
