Abstract:
We 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.

Abstract:
The theory of flat Pseudo-Riemannian manifolds and flat affine manifolds is closely connected to the topic of prehomogeneous affine representations of Lie groups. In this article, we exhibit several aspects of this correspondence. At the heart of our presentation is a development of the theory of characteristic classes and characters of prehomogeneous affine representations. We give applications concerning flat affine, as well as Pseudo-Riemannian and symplectic affine flat manifolds.

Abstract:
We develop the foundations of the deformation theory of compact complete affine space forms and affine crystallographic groups. Using methods from the theory of linear algebraic groups we show that these deformation spaces inherit an algebraic structure from the space of crystallographic homomorphisms. We also study the properties of the action of the homotopy mapping class groups on deformation spaces. In our context these groups are arithmetic groups, and we construct examples of flat affine manifolds where every finite group of mapping classes admits a fixed point on the deformation space. We also show that the existence of fixed points on the deformation space is equivalent to the realisation of finite groups of homotopy equivalences by finite groups of affine diffeomorphisms. Extending ideas of Auslander we relate the deformation spaces of affine space forms with solvable fundamental group to deformation spaces of manifolds with nilpotent fundamental group. We give applications concerning the classification problem for affine space forms.

Abstract:
The group action which defines the moduli problem for the deformation space of flat affine structures on the two-torus is the action of the affine group $\Aff(2)$ on $\bbR^2$. Since this action has non-compact stabiliser $\GL(2,\bbR)$, the underlying locally homogeneous geometry is highly non-Riemannian. In this article, we describe the deformation space of all flat affine structures on the two-torus. In this context interesting phenomena arise in the topology of the deformation space, which, for example, is \emph{not} a Hausdorff space. This contrasts with the case of constant curvature metrics, or conformal structures on surfaces, which are encountered in classical Teichm\"uller theory. As our main result on the space of deformations of flat affine structures on the two-torus we prove that the holonomy map from the deformation space to the variety of conjugacy classes of homomorphisms from the fundamental group of the two-torus to the affine group is a local homeomorphism.

Abstract:
We show that the outer automorphism group of a polycyclic-by-finite group is an arithmetic group. This result follows from a detailed structural analysis of the automorphism groups of such groups. We use an extended version of the theory of the algebraic hull functor initiated by Mostow. We thus make applicable refined methods from the theory of algebraic and arithmetic groups. We also construct examples of polycyclic-by-finite groups which have an automorphism group which does not contain an arithmetic group of finite index. Finally we discuss applications of our results to the groups of homotopy self-equivalences of K(\Gamma, 1)-spaces and obtain an extension of arithmeticity results of Sullivan in rational homotopy theory.

Abstract:
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.

Abstract:
We give a purely geometrical smooth characterization of closed infrasolv manifolds and orbifolds by showing that, up to diffeomorphism, these are precisely the spaces which admit a collapse with bounded curvature and diameter to compact flat orbifolds. Moreover, we distinguish irreducible smooth fake tori geometrically from standard ones by proving that the former have non-vanishing D-minimal volume.

Abstract:
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 $C(\gsp(\bR^{2n}))$.

Abstract:
We characterise the virtually abelian groups which are fundamental groups of compact K\"ahler manifolds and of smooth projective varieties. We show that a virtually abelian group is K\"ahler if and only if it is projective. In particular, this allows to describe the K\"ahler condition for such groups in terms of integral symplectic representations.

Abstract:
Let $G$ be a simply connected, solvable Lie group and $\Gamma$ a lattice in $G$. The deformation space $\mathcal{D}(\Gamma,G)$ is the orbit space associated to the action of $\Aut(G)$ on the space $\mathcal{X}(\Gamma,G)$ of all lattice embeddings of $\Gamma$ into $G$. Our main result generalises the classical rigidity theorems of Mal'tsev and Sait\^o for lattices in nilpotent Lie groups and in solvable Lie groups of real type. We prove that the deformation space of every Zariski-dense lattice $\Gamma$ in $G$ is finite and Hausdorff, provided that the maximal nilpotent normal subgroup of $G$ is connected. This implies that every lattice in a solvable Lie group virtually embeds as a Zariski-dense lattice with finite deformation space. We give examples of solvable Lie groups $G$ which admit Zariski-dense lattices $\Gamma$ such that $\mathcal{D}(\Gamma,G)$ is countably infinite, and also examples where the maximal nilpotent normal subgroup of $G$ is connected and simultaneously $G$ has lattices with uncountable deformation space.