arXiv subjects

Oliver Baues

Publications and source records attributed to Oliver Baues.

At least 19 recordsLinked to original sources

arXiv2023

On the automorphism group of parabolic structures and closed aspherical manifolds

In this expository paper we discuss several properties on closed aspherical parabolic ${\sfG}$-manifolds $X/Γ$. These are manifolds $X/Γ$, where $X$ is a smooth contractible manifold with a parabolic ${\sfG}$-structure for which $Γ\leq \Aut_{\sfG}(X)$ is a discrete subgroup acting properly discontinuously on $X$ with compact quotient. By a parabolic $\sfG$-structure on $X$ we have in mind a Cartan structure which is modeled on one of the classical parabolic geometries arising from simple Lie groups $\sfG$ of rank one. Our results concern in particular the properties of the automorphism groups $\Aut_{\sfG}(X/Γ)$. Our main results show that the existence of certain parabolic ${\sfG}$-structures can pose strong restrictions on the topology of compact aspherical manifolds $X/Γ$ and their parabolic automorphism groups. In this realm we prove that any compact aspherical standard $CR$-manifold with virtually solvable fundamental group is diffeomorphic to a quotient of a Heisenberg manifold of complex type with its standard $CR$-structure. Furthermore we discuss the analogue properties of standard quaternionic contact manifolds in relation to the quaternionic Heisenberg group.

math.DG↗

arXiv2021

A note on vanishing of equivariant differentiable cohomology of proper actions and application to CR-automorphism and conformal groups

We establish that for any proper action of a Lie group on a manifold the associated equivariant differentiable cohomology groups with coefficients in modules of $\mathcal{C}^\infty$-functions vanish in all degrees except than zero. Furthermore let $G$ be a Lie group of $CR$-automorphisms of a strictly pseudo-convex $CR$-manifold $M$. We associate to $G$ a canonical class in the first differential cohomology of $G$ with coefficients in the $\mathcal{C}^\infty$-functions on $M$. This class is non-zero if and only if $G$ is essential in the sense that there does not exist a $CR$-compatible strictly pseudo-convex pseudo-Hermitian structure on $M$ which is preserved by $G$. We prove that a closed Lie subgroup $G$ of $CR$-automorphisms acts properly on $M$ if and only if its canonical class vanishes. As a consequence of Schoen's theorem, it follows that for any strictly pseudo-convex $CR$-manifold $M$, there exists a compatible strictly pseudo-convex pseudo-Hermitian structure such that the CR-automorphism group for $M$ and the group of pseudo-Hermitian transformations coincide, except for two kinds of spherical $CR$-manifolds. Similar results hold for conformal Riemannian and Kähler manifolds.

math.DG↗

arXiv2020

Rigidity of pseudo-Hermitian homogeneous spaces of finite volume

Let $M$ be a pseudo-Hermitian homogeneous space of finite volume. We show that $M$ is compact and the identity component $G$ of the group of holomorphic isometries of $M$ is compact. If $M$ is simply connected, then even the full group of holomorphic isometries is compact. These results stem from a careful analysis of the Tits fibration of $M$, which is shown to have a torus as its fiber. The proof builds on foundational results on the automorphisms groups of compact almost pseudo-Hermitian homogeneous spaces. It is known that a compact homogeneous pseudo-Kähler manifold splits as a product of a complex torus and a rational homogeneous variety, according to the Levi decomposition of $G$. Examples show that compact homogeneous pseudo-Hermitian manifolds in general do not split in this way.

math.DG↗

arXiv2019

Isometry Lie algebras of indefinite homogeneous spaces of finite volume

Let $\mathfrak{g}$ be a real finite-dimensional Lie algebra equipped with a symmetric bilinear form $\langle\cdot,\cdot\rangle$. We assume that $\langle\cdot,\cdot\rangle $ is nil-invariant. This means that every nilpotent operator in the smallest algebraic Lie subalgebra of endomomorphims containing the adjoint representation of $\mathfrak{g}$ is an infinitesimal isometry for $\langle\cdot,\cdot\rangle $. 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 $\mathfrak{g} $ act by infinitesimal isometries for $\langle\cdot,\cdot\rangle $. Moreover, we study properties of the kernel of $\langle\cdot,\cdot\rangle $ and the totally isotropic ideals in $\mathfrak{g} $ in relation to the index of $\langle\cdot,\cdot\rangle $. 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 two. Examples show that the theory becomes significantly more complicated for index greater than two.

math.DG↗

arXiv2019

Locally Homogeneous Aspherical Sasaki Manifolds

Let $G/H$ be a contractible homogeneous Sasaki manifold. A compact locally homogeneous aspherical Sasaki manifold $Γ\big\backslash G/H$ is by definition a quotient of $G/H$ by a discrete uniform subgroup $Γ\leq G$. We show that a compact locally homogeneous aspherical Sasaki manifold is always quasi-regular, that is, $Γ\big\backslash G/H$ is an $S^{1}$-Seifert bundle over a locally homogeneous aspherical Kähler orbifold. We discuss the structure of the isometry group $\mathrm{Isom}(G/H)$ for a Sasaki metric of $G/H$ in relation with the pseudo-Hermitian group $\mathrm{Psh} (G/H)$ for the Sasaki structure of $G/H$. We show that a Sasaki Lie group $G$, when $Γ\big\backslash G$ 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. In addition, we classify all aspherical Sasaki homogeneous spaces for semisimple Lie groups.

math.DG↗

arXiv2018

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.

math.DG↗

arXiv2018

Simply connected indefinite homogeneous spaces of finite volume

Let $M$ be a simply connected pseudo-Riemannian homogeneous space of finite volume with isometry group $G$. We show that $M$ is compact and that the solvable radical of $G$ is abelian and the Levi factor is a compact semisimple Lie group acting transitively on $M$. For metric index less than three, we find that the isometry group of $M$ is compact itself. Examples demonstrate that $G$ is not necessarily compact for higher indices. To prepare these results, we study Lie algebras with abelian solvable radical and a nil-invariant symmetric bilinear form. For these, we derive an orthogonal decomposition into three distinct types of metric Lie algebras.

math.DG↗

arXiv2015

Rigidity of compact pseudo-Riemannian homogeneous spaces for solvable Lie groups

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 on $M$ is induced by a bi-invariant pseudo-Riemannian metric on $G$. Furthermore, we show that the identity component of the isometry group of $M$ coincides with $G$.

math.DG↗

arXiv2013

Symplectic Lie Groups I-III

We develop the structure theory of symplectic Lie groups based on the study of their isotropic normal subgroups. The article consists of three main parts. In the first part we show that every symplectic Lie group admits a sequence of subsequent symplectic reductions to a unique irreducible symplectic Lie group. The second part concerns the symplectic geometry of cotangent symplectic Lie groups and the theory of Lagrangian extensions of flat Lie groups. In the third part of the article we analyze the existence problem for Lagrangian normal subgroups in nilpotent symplectic Lie groups.

math.DG↗

arXiv2012

Seifert fiberings and collapsing of infrasolv spaces

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.

math.DG↗

arXiv2011

The deformations of flat affine structures on the two-torus

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 Teichmüller 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.

math.DG↗

arXiv2011

Deformations and rigidity of lattices in solvable Lie groups

Let $G$ be a simply connected, solvable Lie group and $Γ$ a lattice in $G$. The deformation space $\mathcal{D}(Γ,G)$ is the orbit space associated to the action of $\Aut(G)$ on the space $\mathcal{X}(Γ,G)$ of all lattice embeddings of $Γ$ into $G$. Our main result generalises the classical rigidity theorems of Mal'tsev and Saitô 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 $Γ$ 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 $Γ$ such that $\mathcal{D}(Γ,G)$ 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.

math.DG↗

arXiv2010

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.

math.DG↗

arXiv2009

Virtually abelian Kähler and projective groups

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.

math.DG↗

arXiv2008

Prehomogeneous Affine Representations and Flat Pseudo-Riemannian Manifolds

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.

math.DG↗

arXiv2008

Deformation Spaces for Affine Crystallographic Groups

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.

math.DG↗

arXiv2006

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 $\bbC^n$ by a discrete group of complex isometries.

math.DG↗

arXiv2005

Automorphism groups of polycyclic-by-finite groups and arithmetic groups

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.

math.GR↗