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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 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]

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. [Copyright &y& Elsevier]

Let Γ be a virtually polycyclic group so that the Fitting subgroup is torsion-free and contains its centralizer. We prove that an effective extension of Γ by a finite group μ is isomorphic to an affine crystallographic group if and only if there exists a fixed point for the action of μ on the deformation space of affine crystallographic actions of Γ. We associate to Γ a finitely generated torsion-free nilpotent group Θ which is called the unipotent shadow of Γ, and we relate the deformation space of Γ to the deformation space of Θ. As an application, we show that Γ is isomorphic to an affine crystallographic group if, e.g., Θ has nilpotency class &les;3, or if the polycylic rank of Γ is &les;5, and also in some other cases. To cite this article: O. Baues, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 785–788. [Copyright &y& Elsevier]

Let G / H be a contractible homogeneous Sasaki manifold. A compact locally homogeneous aspherical Sasaki manifold Γ \ G / H is by definition a quotient of G / H by a discrete uniform subgroup Γ ≤ G. We show that a compact locally homogeneous aspherical Sasaki manifold is always quasi-regular, that is, Γ \ G / H is an S 1 -Seifert bundle over a locally homogeneous aspherical Kähler orbifold. We discuss the structure of the isometry group Isom (G / H) for a Sasaki metric of G / H in relation with the pseudo-Hermitian group Psh (G / H) for the Sasaki structure of G / H. We show that a Sasaki Lie group G , when Γ \ G is a compact locally homogeneous aspherical Sasaki manifold, is either the universal covering group of SL (2 , R) or a modification of a Heisenberg nilpotent Lie group with its natural Sasaki structure. We also show that any compact regular aspherical Sasaki manifold with solvable fundamental group is finitely covered by a Heisenberg manifold and its Sasaki structure may be deformed to a locally homogeneous one. In addition, we classify all aspherical Sasaki homogeneous spaces for semisimple Lie groups. [ABSTRACT FROM AUTHOR]