Volume of representations and mapping degree

Given a connected real Lie group and a contractible homogeneous proper G–space X furnished with a G–invariant volume form, a real valued volume can be assigned to any representation ρ:π1(M)→G for any oriented closed smooth manifold M of the same dimension as X. Suppose that G contains a closed and cocompact semisimple subgroup, it is shown in this paper that the set of volumes is finite for any given M. From a perspective of model geometries, examples are investigated and applications with mapping degrees are discussed.