  We develop a theory of large scale geometry of metrisable topological groups
that, in a significant number of cases, allows one to define and identify a
unique quasi-isometry type intrinsic to the topological group. Moreover, this
quasi-isometry type coincides with the classical notion in the case of
compactly generated locally compact groups and, for the additive group of a
Banach space, is simply that of the corresponding Banach space. In particular,
we characterise the class of separable metrisable groups admitting metrically
proper, respectively, maximal compatible left-invariant metrics. Moreover, we
develop criteria for when a metrisable group admits metrically proper affine
isometric actions on Banach spaces of various degress of convexity and
reflexivity.
  A further study of the large scale geometry of automorphism groups of
countable first order model theoretical structures is separated into a
companion paper.
