Manifolds with infinitely many actions of an arithmetic group (Q919325)

Cocompact discrete subgroups \(\Gamma\) of SL(n,\({\mathbb{R}})\) and SU(p,q) together with manifolds M are described such that there are infinitely many \(\Gamma\)-actions on M mutually not conjugate in Diff(M), Homeo(M) and Meas(M). Here Meas(M) is the group of measure class preserving automorphisms. Each such action leaves a smooth metric invariant and every \(\Gamma\) orbit is dense. This is proved - embedding \(\Gamma\)- from: Thm.2: There is a family of non-equivalent tori \(T_ k<SU(n)\times SU(n)\) with \(SU(n)\times SU(n)/T_ k\) all diffeomorphic. Choosing the \(T_ k\) as graphs over a torus T in the first SU(n)-factor, conjugacy can be checked easily. The assertion then follows from a sufficiently fine classification of the SU(n)-bundles \(SU(n)\times SU(n)/T_ k\to SU(n)/T\).