Lie groups with a small space of metric structures
In this talk we will consider a family of solvable, non-nilpotent Lie groups, including the three-dimensional group SOL. On such a group, any pair of left-invariant Riemannian metrics are found to be roughly similar: after multiplying one of them by a suitable multiplicative constant, they will differ by at most a bounded amount. This allows one to reformulate various earlier results about the quasi isometries of these groups in a common framework.
I will compare this result with a recent theorem of Oregon-Reyes, giving an opposite conclusion when considering non-elementary word-hyperbolic groups: the latter are found to have large spaces of metric structures.
Joint work with Enrico Le Donne and Xiangdong Xie.