  Megrelishvili defines \emph{light groups} of isomorphisms of a Banach space
as the groups on which the Weak and Strong Operator Topologies coincide, and
proves that every bounded group of isomorphisms of Banach spaces with the Point
of Continuity Property (PCP) is light. We investigate this concept for
isomorphism groups $G$ of classical Banach spaces $X$ without the PCP,
specially isometry groups, and relate it to the existence of $G$-invariant LUR
or strictly convex renormings of $X$.
