Groups as abstract structures were recognized progressively during the 19th century by mathematicians including Gauss (Disquisitiones avithmeticae in
1801), Cauchy, Galois, Cayley, Jordan, Sylow, Frobenius, Klein (Erlangen pro-
gram in 1872), Lie, Poincar6 ... ; see e.g. Chapter III in [Dieud-78] and [NeumP-
99]. Groups are of course sets given with appropriate "multiplications", and
they are often given together with actions on interesting geometric objects.
But the fact which we want to stress here is that groups are also interesting
geometric objects by themselves — a point of view illustrated in the past by
Cayley and Dehn (see Chapter 1.5 in [ChaMa-82]), and more recently by Gro-
mov. More precisely, a finitely-generated group can be seen as a metric space
(the distance between two points being defined "up to quasi-isometry", in the
sense of Section IV.B below), and this gives rise to a very fruitful approach to
group theory.