Characteristic Classes and Representations of Discrete Subgroups of Lie Groups

A volume invariant is used to characterize those representations of a countable group into a connected semisimple Lie group G which are injective and whose image is a discrete cocompact subgroup of G. Let IT be a discrete cocompact subgroup of G and consider the analytic variety Hom(7r, G) consisting of homomorphisms 0: TT —> G. Denote by K a maximal compact subgroup of G and X = K\G the associated symmetric space. Let M be the orbit space X/TT. (For convenience we shall henceforth assume that n is torsionfree: by Selberg's lemma [12] this may be accomplished by replacing TT by a subgroup of finite index. This insures that M is a compact smooth manifold having TT as its fundamental group. The case when n has torsion follows from the torsionfree case with minor modifications but these modifications need not concern us here.) To every representation 0 G Hom(7r, G) we associate a foliated bundle E^ over M with fibre X and structure group G (see e.g. [6] ). If co is a closed Ginvariant differential fc-form on X then we may spread co over the fibres of E^ (copies of X) to obtain a closed fc-form co^ on E^. We define co(0) = fj^f*^^ where ƒ is any section of E^. Moreover co(0) is independent of the choice of section. For example taking co to be the G-invariant volume form on X we obtain a real number u(0) which depends on 0. When X is even-dimensional the Chern-Gauss-Bonnet theorem implies that u(0) may be described as an Euler number, i.e. the self-intersection number of any section, which is a topological invariant of E^. When X is odd-dimensional this volume invariant is related to a more recent kind of topological invariant (based on bounded cohomology and due to Gromov [3] ) and is constant on the connected components of Hom(7r, G). The volume invariant satisfies an inequality