The geodesic growth for virtually abelian groups (Alex Bishop, UNIGE)

04.10.2022 10:30

Let X be a finite generating set for a group. The volume growth counts the number of group elements which can be spelled out by words of a given length, in the letters X. The geodesic growth function counts the number of minimal-length spellings of elements, i.e., the number of geodesic words of a given length.

It is a result of Benson (1983) that the volume growth for a virtually abelian group is rational with respect to every generating set. In this talk, we show an analogous result for geodesic growth. In particular, we show that the geodesic growth of any virtually abelian group is D-finite with respect to every generating set. We prove this result by modifying the techniques of Benson to show that the language of geodesics belongs to the family of RCM languages introduced by Castiglione and Massazza (2017).

Lieu

Bâtiment: Conseil Général 7-9

Room 1-05, Tuesday 4.10.2022, Séminaire "Groupes et géométrie"

entrée libre