Intermediate geodesic growth in virtually nilpotent groups (Corentin Bodart, UNIGE)

09.05.2023 10:30

Geodesic growth counts the number of geodesics of length n in the Cayley graph of a group G (with a generating set S). As soon as the group has exponential volume growth, the geodesic growth is also exponential. Finding pairs (G,S) with polynomial geodesic growth is trickier but, by now, several constructions are known. At last, a question that has been around and open since the 90's is the existence of a pair (G,S) with intermediate geodesic growth. Most of the efforts to construct such an example have been centered on groups of intermediate volume growth, without success. Perhaps surprisingly, we show that intermediate geodesic growth is possible in the realm of virtually nilpotent groups.

In this talk, I will introduce our main example, a virtually 3-step nilpotent group, and a geometric model for it. I will explain the main ideas necessary for both the upper and lower bounds on the number of geodesics, and compare them with previous arguments of Shapiro, Bridson-Burillo-Elder-Sunic and Bishop-Elder.

Lieu

Bâtiment: Conseil Général 7-9

Room 1-05, Séminaire "Groupes et géométrie"

entrée libre