Relative amenability

We introduce a relative fixed point property for subgroups of a locally compact group, which we call relative amenability . It is a priori weaker than amenability. We establish equivalent conditions, related among others to a problem studied by Reiter in 1968. We record a solution to Reiter's problem. We study the class \mathscr{X} of groups in which relative amenability is equivalent to amenability for all closed subgroups; we prove that \mathscr{X} contains all familiar groups. Actually, no group is known to lie outside \mathscr{X} . Since relative amenability is closed under Chabauty limits, it follows that any Chabauty limit of amenable subgroups remains amenable if the ambient group belongs to the vast class \mathscr{X} .