A class of Sasakian 5-manifolds

We obtain some general results on Sasakian Lie algebras and prove as a consequence that a (2n + 1)-dimensional nilpotent Lie group admitting left-invariant Sasakian structures is isomorphic to the real Heisenberg group H _{2n + 1}. Furthermore, we classify Sasakian Lie algebras of dimension five and determine which of them carries a Sasakian α-Einstein structure. We show that a five-dimensional solvable Lie group with a left-invariant Sasakian structure and which admits a compact quotient by a discrete subgroup is isomorphic to either H ₅ or a semidirect product ? ? (H ₃ × ?). In particular, the compact quotient is an S ¹-bundle over a four-dimensional Kähler solvmanifold.     

Assouad–Nagata dimension of connected Lie groups

We prove that the asymptotic Assouad–Nagata dimension of a connected Lie group G equipped with a left-invariant Riemannian metric coincides with its topological dimension of G/C where C is a maximal compact subgroup. To prove it we will compute the Assouad–Nagata dimension of connected solvable Lie groups and semisimple Lie groups. As a consequence we show that the asymptotic Assouad–Nagata dimension of a polycyclic group equipped with a word metric is equal to its Hirsch length and that some wreath-type finitely generated groups can not be quasi-isometrically embedded into any cocompact lattice on a connected Lie group.     

Product decompositions of solvable Lie groups

Summary The article investigate the structure of real solvable connected Lie groups. It is described how one can decompose a solvable Lie group in direct and semidirect products of closed connected subgroups. In particular, the commutator group, Cartan subgroups, the center, maximal compactly embedded subgroups and tori are considered. Furthermore, one can find special solvable Lie groups and their product decompositions, namely compactly generated solvable Lie groups and those Lie groups which are generated by maximal compactly embedded subgroups. This article was processed by the author using the LATEX style filecljour1 from Springer-Verlag.     

Symplectic half-flat solvmanifolds

We classify solvable Lie groups admitting left invariant symplectic half-flat structure. When the Lie group has a compact quotient by a lattice, we show that these structures provide solutions of supersymmetric equations of type IIA.     

Minimality and unique ergodicity of homogeneous actions

Consider a connected Lie groupG, a lattice Γ inG, a connected subgroupH ofG, and the adjoint representation Ad ofG on its Lie algebra g. Suppose that Ad(H) splits into a semidirect product of a reductive subgroup and the unipotent radical. We prove that the minimality of the leftH-action onG/Γ then implies its unique ergodicity. Simultaneously, we suggest a reduction of the study of finite ergodic measures for an arbitrary action (G/Γ,H), where the subgroupH∈G is connected and Γ∈G is discrete, to the case of an Abelian subgroupH. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 293–301, August, 1999.     

The structure of abelian pro-Lie groups

A pro-Lie group is a projective limit of a projective system of finite dimensional Lie groups. A prodiscrete group is a complete abelian topological group in which the open normal subgroups form a basis of the filter of identity neighborhoods. It is shown here that an abelian pro-Lie group is a product of (in general infinitely many) copies of the additive topological group of reals and of an abelian pro-Lie group of a special type; this last factor has a compact connected component, and a characteristic closed subgroup which is a union of all compact subgroups; the factor group modulo this subgroup is pro-discrete and free of nonsingleton compact subgroups. Accordingly, a connected abelian pro-Lie group is a product of a family of copies of the reals and a compact connected abelian group. A topological group is called compactly generated if it is algebraically generated by a compact subset, and a group is called almost connected if the factor group modulo its identity component is compact. It is further shown that a compactly generated abelian pro-Lie group has a characteristic almost connected locally compact subgroup which is a product of a finite number of copies of the reals and a compact abelian group such that the factor group modulo this characteristic subgroup is a compactly generated prodiscrete group without nontrivial compact subgroups.Mathematics Subject Classification (1991): 22B, 22E     

Two-step solvable Lie algebras and weight graphs

By using the concept of weight graph associated to nonsplit complex nilpotent Lie algebras \mathfrakg\mathfrak{g}, we find necessary and sufficient conditions for a semidirect product \mathfrakg?^? T_i\mathfrak{g}\overrightarrow{\oplus } T_{i} to be two-step solvable, where $T_{i}TT over \mathfrakg\mathfrak{g} which induces a decomposition of \mathfrakg\mathfrak{g} into one-dimensional weight spaces without zero weights. In particular we show that the semidirect product of such a Lie algebra with a maximal torus of derivations cannot be itself two-step solvable. We also obtain some applications to rigid Lie algebras, as a geometrical proof of the nonexistence of two-step nonsplit solvable rigid Lie algebras in dimensions n\geqslant 3n\geqslant 3.     

Global rigidity of holomorphic Riemannian metrics on compact complex 3-manifolds

We study compact complex 3-manifolds M admitting a (locally homogeneous) holomorphic Riemannian metric g. We prove the following: (i) If the Killing Lie algebra of g has a non trivial semi-simple part, then it preserves some holomorphic Riemannian metric on M with constant sectional curvature; (ii) If the Killing Lie algebra of g is solvable, then, up to a finite unramified cover, M is a quotient Γ\G, where Γ is a lattice in G and G is either the complex Heisenberg group, or the complex SOL group. S. Dumitrescu was partially supported by the ANR Grant BLAN 06-3-137237.     

On the cohomology of homogeneous spaces of finite loop spaces and the Eilenberg–Moore spectral sequence

We prove a collapse theorem for the Eilenberg–Moore spectral sequence and as an application we show that under certain conditions the cohomology of a homogeneous space of a connected finite loop space with a maximal rank torsion free subgroup is concentrated in even degrees and torsionfree, generalizing classical theorems for compact Lie groups of Borel and Bott.     

Geometric lattice actions, entropy and fundamental groups

Let be a lattice in a noncompact simple Lie Group G, where . Suppose acts analytically and ergodically on a compact manifold M preserving a unimodular rigid geometric structure (e.g. a connection and a volume). We show that either the action is isometric or there exists a "large image" linear representation of . Under an additional assumption on the dynamics of the action, we associate to a virtual arithmetic quotient of full entropy. Received: December 14, 2000     

Zusammenhängende Untergruppen von pro-Lieschen Gruppen

In this paper we consider the lattice G of all closed connected subgroups of pro-Lie groups G, which seems to have in some sense a more geometric nature than the full lattice of all closed subgroups. We determine those pro-Lie groups whose lattice shares one of the elementary geometric lattice properties, such as the existence of complements and relative complements, semi-modularity and its dual, the chain condition, self-duality and related ones. Apart from these results dealing with subgroup lattices we also get two structure theorems, one saying that maximal closed analytic subgroups of Lie groups actually are maximal among all analytic subgroups, the other that each connected abelian pro-Lie group is a direct product of a compact group with copies of the reals.     

Aspherical Kähler manifolds with solvable fundamental group

We survey recent developments which led to the proof of the Benson-Gordon conjecture on Kähler quotients of solvable Lie groups. In addition, we prove that the Albanese morphism of a Kähler manifold which is a homotopy torus is a biholomorphic map. The latter result then implies the classification of compact aspherical Kähler manifolds with (virtually) solvable fundamental group up to biholomorphic equivalence. They are all biholomorphic to complex manifolds which are obtained as a quotient of $\mathbb{C}^{n}We survey recent developments which led to the proof of the Benson-Gordon conjecture on K?hler quotients of solvable Lie groups. In addition, we prove that the Albanese morphism of a K?hler manifold which is a homotopy torus is a biholomorphic map. The latter result then implies the classification of compact aspherical K?hler manifolds with (virtually) solvable fundamental group up to biholomorphic equivalence. They are all biholomorphic to complex manifolds which are obtained as a quotient of \mathbbCⁿ\mathbb{C}^{n} by a discrete group of complex isometries.     

Some subgroups of the full linear group

Further observations are made on the author's earlier paper (Ref. Zh. Mat., 1977, 5A284) dealing with the lattice H of all subgroups of the full linear group GL(n, K) over a field K that contain the group K of diagonal matrices. It is noted, for example, that for an infinite field K all subgroups inD(n, K) are algebraic; a subgroup in H is connected if and only if it is a net subgroup; the lattice of all connected subgroups in H is isomorphic to the lattice of all marked topologies onn points; any subgroup H in H is a semidirect product H=A·H_o of a maximal connected normal subgroup H_o of H and a finite group A of, permutation matrices.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 71, pp. 42–46, 1977.     

On the symmetry of L1 algebras of locally compact motion groups,and the Wiener Tauberian theorem

Let G be a locally compact motion group, i.e., it is a semidirect product of a compact subgroup with a closed abelian normal subgroup, the action of the compact subgroup on the other one being by conjugation. The main result of this paper is that the group algebra of such a group is symmetric. This result is then used to prove that a generalization of the Wiener-Tauberian theorem holds for such groups. Precisely, it is shown that every proper closed two-sided ideal in L₁(G) is annihilated by an irreducible unitary representation of G, lifted to L₁(G).     

On the lower weakly solvable l-radical of lattice ordered lie algebras

The concept of the lower weakly solvable radical of Lie algebras is important in the study of Lie algebras. The purpose of this paper is to investigate the generalization of this concept to lattice ordered Lie algebras over partially ordered fields. Some results concerning properties of the lower weakly solvable l-radical of lattice ordered Lie algebras are obtained. Necessary and sufficient conditions for the l-prime radical of a Lie l-algebra to be equal to the lower weakly solvable l-radical of the Lie l-algebra are presented.     

Kernels of maps between classifying spaces

For homomorphisms between groups, one can divide out the kernel to get an injection. Here, we develop a notion of kernels for maps between classifying spaces of compact Lie groups. We show that the kernel is a normal subgroup in a modified sense and prove a generalization of a theorem of Quillen, namely, a mapf:BG→BH _p ^∧ is injective, iff the induced map in mod-p cohomology is finite. Moreover, for compact connected Lie groups, every mapf:BG→BH _p ^∧ factors over a quotient ofG in a modified sense and this factorisation is an injection.     

Factoring a Lie Group into a Compactly Embeddedand a Solvable Subgroup

 Let G be a real connected Lie group. A subgroup K is called compactly embedded if the closure of Ad(K) is compact in Aut(). If K is, in addition, maximal with respect to this property, then there exists a solvable subgroup S containing the nilradical such that and is the one-component of the center of G. (Received 1 June 1999; in revised form 28 December 1999)     

On the Euler characteristic of definable groups

We show that in an arbitrary o‐minimal structure the following are equivalent: (i) conjugates of a definable subgroup of a definably connected, definably compact definable group cover the group if the o‐minimal Euler characteristic of the quotient is non zero; (ii) every infinite, definably connected, definably compact definable group has a non trivial torsion point (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Brody quotients of homogeneous Stein manifolds

Homogeneous Stein manifolds admit a complex equivariant quotient with respect to Brody hyperbolicity. As a consequence, simply connected Stein manifolds homogeneous by a solvable Lie group split into a product of a bounded homogenous Stein domain and a complex cell.