The metric geometry of infinite dimensional Lie groups and their homogeneous spaces

We study the geometry of Lie groups $G$ with a continuous Finsler metric, assuming the existence of a subgroup $K$ such that the metric is right-invariant for the action of $K$. We present a systematic study of the metric and geodesic structure of homogeneous spaces $M$ obtained by the quotient $M\simeq G\mathrm{/}K$. Of particular interest are left-invariant metrics of $G$ which are then bi-invariant for the action of $K$. We then focus on the geodesic structure of groups $K$ that admit bi-invariant metrics, proving that one-parameter groups are short paths for those metrics, and characterizing all other short paths. We provide applications of the results obtained, in two settings: manifolds of Banach space linear operators, and groups of maps from compact manifolds.