Representations of Fundamental Groups of Manifolds with a Semisimple Transformation Group

In this paper, we examine the relationship between the topology of a manifold M , specifically the finite dimensional representation theory of the fundamental group 7t, (M), and the Lie groups that can act on M. More precisely, let G be a connected semisimple Lie group of higher real rank, and suppose G acts continuously on a (topological) manifold M, preserving a finite measure. The main theme of this paper is that the representation theory of 7t, (M) in low dimensions is to a large extent controlled by that of G (the latter of course being well understood). In particular, under natural hypotheses (e.g., that the action of G on M is engaging, i.e., there is no loss of ergodicity in passing to finite covers; see Definition 3.1 below), we prove that if G has no nontrivial representations below dimension d, then every representation of 1t, (M) below dimension d is finite; that is, it factors through a finite quotient group. Under different but related hypotheses (namely that the action is topologically engaging, that is, roughly speaking, that the action is proper on the universal cover; see Definition 3.2), we show that 7t, (M) admits no faithful representation over Q below dimension d. These results of course impose severe restrictions on the manifolds on which G can act. The hypotheses of engaging or topological engaging are quite mild, and one or the other is satisfied in every known nontrivial example. We also remark that Gromov has shown [5] that every real analytic connection preserving action of G is tOpologically engaging. Rather than considering representations, one can consider, more generally, homomorphisms into a general algebraic group, and the above results become special cases of the following theorems. (These results, as well as most of the others in this paper, hold for spaces much more general than the class of topological manifolds. See the beginning of §2.)