  Usual gauge fixing procedures in classical general relativity rely on the
existence of solutions of a second order wave equation. We propose to use the
equation to relate asymptotic symmetries at infinity to asymptotic symmetries
of a black hole horizon, in tune with recent proposals. We illustrate the
construction for the BTZ and four-dimensional Kerr black holes. We find in both
cases a realization of the group of diffeomorphisms of the circle $S^1$.
