  We introduce the concept of an $\mathrm{L}^{1}$ full group associated with a
measure-preserving action of a Polish normed group on a standard probability
space. Such groups are shown to carry a natural separable complete metric, and
are thus Polish. Our construction generalizes $\mathrm{L}^{1}$ full groups of
actions of discrete groups, which have been studied recently by the first
author.
  We show that under minor assumptions on the actions, topological derived
subgroups of $\mathrm{L}^{1}$ full groups are topologically simple and -- when
the acting group is locally compact and amenable -- are whirly amenable and
generically two-generated. $\mathrm{L}^{1}$ full groups of actions of compactly
generated locally compact Polish groups are shown to remember the
$\mathrm{L}^{1}$ orbit equivalence class of the action.
  For measure-preserving actions of the real line (also often called
measure-preserving flows), the topological derived subgroup of an
$\mathrm{L}^{1}$ full groups is shown to coincide with the kernel of the index
map, which implies that $\mathrm{L}^{1}$ full groups of free measure-preserving
flows are topologically finitely generated if and only if the flow admits
finitely many ergodic components. The latter is in a striking contrast to the
case of $\mathbb{Z} $-actions, where the number of topological generators is
controlled by the entropy of the action.
  We also study the coarse geometry of the $\mathrm{L}^{1}$ full groups. The
$\mathrm{L}^{1}$ norm on the derived subgroup of the $\mathrm{L}^{1}$ full
group of an aperiodic action of a locally compact amenable group is proved to
be maximal in the sense of Rosendal. For measure-preserving flows, this holds
for the $\mathrm{L}^{1}$ norm on all of the $\mathrm{L}^{1}$ full group.
