  A group of bijections G acting on a set X is said with fixed points
(abbreviated as gaf from the french "groupe {\`a} points fixes") if any element
of G has at least one fixed point in X. The G group is said with a common fixed
point (abbreviated as gag for "groupe {\`a} point fixe global") if there is x
$\in$ X fixed by all elements of G. The group G is said fixating if any
subgroup of G which is a gaf is automatically a gag. The article explores which
groups are fixating. The situation depends on the assumptions made on the group
of bijections and on the support set X. For example the group of isometries of
the Euclidean space R n is fixating for n 3 but not for n 4. The case of
isometries of elliptic and hyperbolic spaces is also considered, as well as
that of isometries of some discrete sets. As we will see, the situation depends
upon the fact whether the so-called median inequality is satisfied in the
ambient space or not. Besides, some of our constructions of nonfixating groups
rely on the existence of free subgroups of certain linear groups.
