We study actions of discrete groups on classifying spaces (or
classifying spaces for proper actions). For instance the hyperbolic
plane is a classifying space for proper actions of the group PSL(2,Z)
(but not of minimal dimension). Such spaces can be used to compute the
cohomology of the group, so we want them to have the lowest possible
dimension. This leads us to the definitons of the (proper) geometric
dimension and the (virtual) cohomological dimension. These two
dimensions are not always equal, we will see it is the case for a
lattice in the group of isometries G of a symmetric space of
non-compact type without Euclidean factors (such a group is a
semisimple Lie group but not necessarily connected). This result has
an important consequence called "dimension rigidity", that is, the two
dimensions are still equal for a group commensurable to a lattice of
G.