  This book offers to study locally compact groups from the point of view of
appropriate metrics that can be defined on them, in other words to study
"Infinite groups as geometric objects", as Gromov writes it in the title of a
famous article. The theme has often been restricted to finitely generated
groups, but it can favorably be played for locally compact groups. The
development of the theory is illustrated by numerous examples, including matrix
groups with entries in the the field of real or complex numbers, or other
locally compact fields such as p-adic fields, isometry groups of various metric
spaces, and, last but not least, discrete group themselves. Word metrics for
compactly generated groups play a major role. In the particular case of
finitely generated groups, they were introduced by Dehn around 1910 in
connection with the Word Problem. Some of the results exposed concern general
locally compact groups, such as criteria for the existence of compatible
metrics on locally compact groups. Other results concern special classes of
groups, for example those mapping onto the group of integers (the Bieri-Strebel
splitting theorem for locally compact groups).
  Prior to their applications to groups, the basic notions of coarse and
large-scale geometry are developed in the general framework of metric spaces.
Coarse geometry is that part of geometry concerning properties of metric spaces
that can be formulated in terms of large distances only. In particular coarse
connectedness, coarse simple connectedness, metric coarse equivalences, and
quasi-isometries of metric spaces are given special attention. The final
chapters are devoted to the more restricted class of compactly presented
groups, generalizing finitely presented groups to the locally compact setting.
They can indeed be characterized as those compactly generated locally compact
groups that are coarsely simply connected.
