  We revisit the boundary dynamics of asymptotically flat, three dimensional
gravity. The boundary is governed by a momentum conservation equation and an
energy conservation equation, which we interpret as fluid equations, following
the membrane paradigm. We reformulate the boundary's equations of motion as
Hamiltonian flow on the dual of an infinite-dimensional, semi-direct product
Lie algebra equipped with a Lie-Poisson bracket. This gives the analogue for
boundary fluid dynamics of the Marsden-Ratiu-Weinstein formulation of the
compressible Euler equations on a manifold, $M$, as Hamiltonian flow on the
dual of the Lie algebra of ${\rm Diff}(M)\ltimes C^\infty(M)$. The Lie group
for boundary fluid dynamics turns out to be ${\rm Diff}(S^1) \ltimes_{\rm Ad}
{\rm \mathfrak{vir}}$, with central charge $c=3/G$. This gives a new derivation
of the centrally extended, three-dimensional Bondi-van der Burg-Metzner-Sachs
(${\rm BMS}_3$) group. The relationship with fluid dynamics helps to streamline
and physically motivate the derivation. For example, the central charge,
$c=3/G$, is simply read off of a fluid equation in much the same way as one
reads off a viscosity coefficient. The perspective presented here may useful
for understanding the still mysterious four-dimensional BMS group.
