We perform a complete and systematic analysis of the solution space of
six-dimensional Einstein gravity. We show that a particular subclass of
solutions -- those that are analytic near I+ -- admit a
non-trivial action of the generalised Bondi-Metzner-van der Burg-Sachs (GBMS)
group which contains \emph{infinite-dimensional} supertranslations and
superrotations. The latter consists of all smooth volume-preserving
Diff×Weyl transformations of the celestial S4. Using the covariant
phase space formalism and a new technique which we develop in this paper (phase
space renormalization), we are able to renormalize the symplectic potential
using counterterms which are \emph{local} and \emph{covariant}. We then
construct charges which faithfully represent the GBMS algebra and in doing so,
settle a long-standing open question regarding the existence of GBMS symmetries
in higher dimensional non-linear gravity. Finally, we show that the
semi-classical Ward identities for the supertranslations and superrotations are
precisely the leading and subleading soft-graviton theorems respectively.Comment: 75 pages, 1 figur