### 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 *d _{1}, d_{2}* on a set

*X*are similar if there is a constant

*λ*>0 such that

*d*for all

_{2}(x,y)=λ d_{1}(x,y)*x, y∈ X*. Two metrics

*d*on a set

_{1}, d_{2}*X*are roughly similar if there are constants

*λ*>0,

*C*≥0 such that

*λd*for all

_{1}(x,y) -C ≤ d_{2}(x,y) ≤ λ d_{1}(x,y)+C*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.

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