Dead ends on wreath products and lamplighter groups (Eduardo Silva, ENS Paris)

28.02.2023 10:30

A finitely generated group G has unbounded depth with respect to a finite generating set S if for any n>=1, there exists g in G such that no element of the form gh, with h of word length at most n, has larger word length than g. In other words, g locally maximizes the word length in its n-neighborhood. The existence of infinite groups with unbounded depth is not evident, and the first known example was the lamplighter group Z/2Z wr Z with standard generators, due to Cleary and Taback.

In this talk we will review what is currently known about depth properties of groups and then concentrate on the case where G=A wr B is the wreath product of two groups. Via the description of geodesics on A wr B in terms of solutions to the traveling salesman problem on B, we are able to prove that for any finite A and finitely generated B, there always exists a standard generating set of A wr B with unbounded depth. We will also exhibit a family of examples which show that the above cannot hold in general for every standard generating set.

Lieu

Bâtiment: Conseil Général 7-9

Room 1-05, Tuesday 28 February 2023, Séminaire "Groupes et géométrie"

entrée libre