Title: Volume of representations and mapping degree,
Abstract: 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 $\rho\colon \pi_1(M)\to G$ 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.