  The complexity of a geodesic language has connections to algebraic properties
of the group. Gilman, Hermiller, Holt, and Rees show that a finitely generated
group is virtually free if and only if its geodesic language is locally
excluding for some finite inverse-closed generating set. The existence of such
a correspondence and the result of Hermiller, Holt, and Rees that finitely
generated abelian groups have piecewise excluding geodesic language for all
finite inverse-closed generating sets motivated our work. We show that a
finitely generated group with piecewise excluding geodesic language need not be
abelian and give a class of infinite non-abelian groups which have piecewise
excluding geodesic languages for certain generating sets. The quaternion group
is shown to be the only non-abelian 2-generator group with piecewise excluding
geodesic language for all finite inverse-closed generating sets. We also show
that there are virtually abelian groups with geodesic languages which are not
piecewise excluding for any finite inverse-closed generating set.
