  We study the Lie group structure of asymptotic symmetry groups in General
Relativity from the viewpoint of infinite-dimensional geometry. To this end, we
review the geometric definition of asymptotic simplicity and emptiness due to
Penrose and the coordinate-wise definition of asymptotic flatness due to Bondi
et al. Then we construct the Lie group structure of the Bondi--Metzner--Sachs
(BMS) group and discuss its Lie theoretic properties. We find that the BMS
group is regular in the sense of Milnor, but not real analytic. This motivates
us to conjecture that it is not locally exponential. Finally, we verify the
Trotter property as well as the commutator property. As an outlook, we comment
on the situation of related asymptotic symmetry groups. In particular, the much
more involved situation of the Newman--Unti group is highlighted, which will be
studied in future work.
