This talk regards the structure of homogeneous spaces in the (bi-)Lipschitz category. Particular attention will be focused on biLipschitz homogeneous geodesic manifolds, i.e., path metric spaces which are homeomorphic to manifolds and have a transitive group of biLipschitz homeomorphisms.
A first result gives a complete description of these manifolds under some additional smoothness assumptions. Namely, if there is a Lie group acting transitively by biLipschitz maps, then such spaces are locally biLipschitz equivalent to homogeneous spaces G/H equipped with a Carnot-Caratheodory metric; here G is a connected Lie group and H is a closed subgroup. This result generalizes a theorem of Berestovskii about isometric transitive actions.
A second result concerns the general two dimensional case. It will be shown that a bi-Lipschitz homogeneous geodesic surface is locally doubling - a property that gives a number of useful metric and measure-theoretic consequences.