Flat Pseudo-Riemannian Homogeneous Spaces with Non-Abelian Holonomy Group

We construct homogeneous flat pseudo-Riemannian manifolds with non-abelian fundamental group. In the compact case, all homogeneous flat pseudo-Riemannian manifolds are complete and have abelian linear holonomy group. To the contrary, we show that there do exist non-compact and non-complete examples, where the linear holonomy is non-abelian, starting in dimensions $\geq 8$, which is the lowest possible dimension. We also construct a complete flat pseudo-Riemannian homogeneous manifold of dimension 14 with non-abelian linear holonomy. Furthermore, we derive a criterion for the properness of the action of an affine transformation group with transitive centralizer

Isometry groups with radical, and aspherical Riemannian manifolds with large symmetry I

Every compact aspherical Riemannian manifold admits a canonical series of orbibundle structures with infrasolv fibers which is called its infrasolv tower. The tower arises from the solvable radicals of isometry group actions on the universal covers. Its length and the geometry of its base measure the degree of continuous symmetry of an aspherical Riemannian manifold. We say that the manifold has large symmetry if it admits an infrasolv tower whose base is a locally homogeneous space. We construct examples of aspherical manifolds with large symmetry, which do not support any locally homogeneous Riemannian metrics

Abelian simply transitive affine groups of symplectic type

We construct a model space C(\gsp(\bR^{2n})) for the variety of Abelian simply transitive groups of affine transformations of type {\rm Sp}(\bR^{2n}). The model is stratified and its principal stratum is a Zariski-open subbundle of a natural vector bundle over the Grassmannian of Lagrangian subspaces in \bR^{2n}. \noindent Next we show that every flat special K\"ahler manifold may be constructed locally from a holomorphic function whose third derivatives satisfy some algebraic constraint. In particular global models for flat special K\"ahler manifolds with constant cubic form correspond to a subvariety of C(\gsp(\bR^{2n})).Comment: corrected typos, updated reference

Infra-solvmanifolds and rigidity of subgroups in solvable linear algebraic groups

AbstractWe give a new proof that compact infra-solvmanifolds with isomorphic fundamental groups are smoothly diffeomorphic. More generally, we prove rigidity results for manifolds which are constructed using affine actions of virtually polycyclic groups on solvable Lie groups. Our results are derived from rigidity properties of subgroups in solvable linear algebraic groups