  The Geroch group is an infinite dimensional transitive group of symmetries of
classical cylindrically symmetric gravitational waves which acts by
non-canonical transformations on the phase space of these waves. Here this
symmetry is rederived and the unique Poisson bracket on the Geroch group which
makes its action on the gravitational phase space Lie-Poisson is obtained. Two
possible notions of asymptotic flatness are proposed that are compatible with
the Poisson bracket on the phase space, and corresponding asymptotic flatness
preserving subgroups of the Geroch group are defined which turn out to be
compatible with the Poisson bracket on the group.
  A quantization of the Geroch group is proposed that is similar to, but
distinct from, the $\mathfrak{sl}_2$ Yangian, and a certain action of this
quantum Geroch group on gravitational observables is shown to preserve the
commutation relations of Korotkin and Samtleben's quantization of
asymptotically flat cylindrically symmetric gravitational waves. The action
also preserves three of the additional conditions that define their
quantization. It is conjectured that the action preserves the remaining two
conditions (asymptotic flatness and a unit determinant condition on a certain
basic field) as well and is, in fact, a symmetry of their model. Our results on
the quantum theory are formal, but a possible rigorous formulation based on
algebraic quantum theory is outlined.
