We want to better understand the structure of metric spaces that are
locally compact, connected and isometrically homogeneous.
After the solution of the Hilbert 5th problem, we know that any such a
space is quasi-isometric to some Lie groups, which can be chosen to be
Moreover, if in addition such spaces are locally connected and of
finite topological dimension, then they are in fact Lie-group
We shall focus on those spaces that are either geodesic metric spaces,
or have polynomial growth, or admit self-similarities.
Respectively, we shall have Carnot groups, quasi-nilpotent groups, and
Joint work with M.Cowling, V.Kivioja, A.Ottazzi, and S.Nicolussi Golo.