On groups whose geodesic growth is polynomial

Abstract

This note records some observations concerning geodesic growth functions. If a nilpotent group is not virtually cyclic then it has exponential geodesic growth with respect to all finite generating sets. On the other hand, if a finitely generated group $G$ has an element whose normal closure is abelian and of finite index, then $G$ has a finite generating set with respect to which the geodesic growth is polynomial (this includes all virtually cyclic groups).

11 pages, 1 figure

References in corpus (2)

• Automatic structures, rational growth and geometrically finite hyperbolic groups
• Regular geodesic languages and the falsification by fellow traveler property

Cited by in corpus (4)

• On the Complexity of the Word Problem for Automaton Semigroups and Automaton Groups
• Periodic Points on Shifts of Finite Type and Commensurability Invariants of Groups
• Geodesic growth in virtually abelian groups
• A virtually 2-step nilpotent group with polynomial geodesic growth