  In this paper we study the possibility of assigning a geometric structure to
the Lie groups. It is shown the Poincar\'{e} and Weyl groups have geometrical
structure of the Riemann-Cartan and Weyl space-time respectively. The geometric
approach to these groups can be carried out by considering the most general
(non)metricity conditions, or equivalently, tetrad postulates which we show
that can be written in terms of the group's gauge fields. By focusing on the
conformal group we apply this procedure to show that a nontrivial 3-metrics
geometry may be extracted from the group's Maurer-Cartan structure equations.
We systematically obtain the general characteristics of this geometry, i.e. its
most general nonmetricity conditions, tetrad postulates and its connections. We
then deal with the gravitational theory associated to the conformal group's
geometry. By proposing an Einstein-Hilbert type action, we conclude that the
resulting gravity theory has the form of quintessence where the scalar field
derivatively coupled to massive gravity building blocks.
