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Let g be a real finite-dimensional Lie algebra equipped with a symmetric bilinear form <.,.>. We assume that <.,.> is nil-invariant. This means that every nilpotent operator in the smallest algebraic Lie subalgebra of endomorphisms containing the adjoint representation of g is an infinitesimal isometry for <.,.>. Among these Lie algebras are the isometry Lie algebras of pseudo- Riemannian manifolds of finite volume. We prove a strong invariance property for nil-invariant symmetric bilinear forms, which states that the adjoint representations of the solvable radical and all simple subalgebras of non-compact type of g act by infinitesimal isometries for <.,.>. Moreover, we study properties of the kernel of <.,.> and the totally isotropic ideals in g in relation to the index of <.,.>. Based on this, we derive a structure theorem and a classification for the isometry algebras of indefinite homogeneous spaces of finite volume with metric index at most 2. Examples show that the theory becomes significantly more complicated for index greater than 2. We apply our results to study simply connected pseudo-Riemannian homogeneous spaces of finite volume. [ABSTRACT FROM AUTHOR]

Let M be a compact connected pseudo-Riemannian manifold on which a solvable connected Lie group G of isometries acts transitively. We show that G acts almost freely on M and that the metric onM is induced by a bi-invariant pseudo-Riemannian metric on G. Furthermore, we show that the identity component of the isometry group ofM coincides with G. The proofs rely on a combination of density properties for uniform subgroups of solvable Lie groups and the reduction theory of pseudo-Riemannian metric Lie groups. [ABSTRACT FROM AUTHOR]

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 ≥ 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 FROM AUTHOR]

We characterise the virtually abelian groups which are fundamental groups of compact Kähler manifolds and of smooth projective varieties. We show that a virtually abelian group is Kähler if and only if it is projective. In particular, this allows to describe the Kähler condition for such groups in terms of integral symplectic representations. [ABSTRACT FROM AUTHOR]

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(Γ,1)-spaces and obtain an extension of arithmeticity results of Sullivan in rational homotopy theory. [ABSTRACT FROM AUTHOR]

We introduce a natural combinatorial curvature function on the corners of plane tessellations and relate it to the global metric geometry of their corresponding edge and dual graphs. If the combinatorial curvature in the corners is non-positive then we prove that any geodesic path in such a graph may be extended to infinity. Moreover, if the combinatorial curvature is negative we show that every pair of geodesic segments with the same end points does not enclose any vertices. We apply these results to establish an estimate for the growth of distance balls, Gromov hyperbolicity, and four-colourability of certain classes of plane tessellations. [ABSTRACT FROM AUTHOR]

Affine structures on surfaces are constructed by gluing polygons. The geometry of the affine surface depends on the shape of the polygon(s) and the particular gluing transformations used. The affine version of the Poincaré fundamental polygon theorem expresses the fundamental group and holonomy of the surface in terms of the gluing data. The theorem may be used to construct all complete affine structures on the 2-torus. The space of inequivalent holonomy representations of such structures is homeomorphic to R2. [ABSTRACT FROM AUTHOR]