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Languages, Geodesics, And HNN Extensions
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. Autostackable groups were introduced by Brittenham, Hermiller, and Holt as a generalization of asynchronously automatic groups on prefix-closed normal forms and groups with finite convergent rewriting systems. Brittenham, Hermiller, and Johnson show that Stallings' non-FP3 group, an HNN extension of a right-angled Artin group, is autostackable. We extend this autostackability result to a larger class of HNN extensions of right-angled Artin groups.
Franke, Maranda, "Languages, Geodesics, And HNN Extensions" (2017). ETD collection for University of Nebraska - Lincoln. AAI10271814.