Groups whose left invariant metrics are roughly similar
Analysis and Geometry Seminar
8th October 2020, 3:15 pm – 4:15 pm
Online, (contact organisers for details)
Two metrics d1, d2 on a set X are similar if there is a constant λ>0 such that d2(x,y)=λ d1(x,y) for all x, y∈ X. Two metrics d1, d2 on a set X are roughly similar if there are constants λ>0, C≥0 such that λd1(x,y) -C ≤ d2(x,y) ≤ λ d1(x,y)+C for all x, y∈ X. In Euclidean spaces there are lots of norms so that the associated metrics are not roughly similar. We exhibit a class of solvable Lie groups (including Heintze groups and SOL type groups) where every two left invariant Riemmanian metrics are roughly similar. This is a new phenomenon. It is open whether any discrete groups (other than virtually cyclic groups) have this property. This is joint work with Enrico Le Donne and Gabriel Pallier.