Related papers: Clifford-Klein forms and a-hyperbolic rank
T. Kobayashi conjectured in the 36th Geometry Symposium in Japan (1989) that a homogeneous space G/H of reductive type does not admit a compact Clifford-Klein form if rank G - rank K < rank H - rank K_H. We solve this conjecture…
A homogeneous space G/H is said to have a compact Clifford-Klein form if there exists a discrete subgroup D of G that acts properly discontinuously on G/H, such that the quotient space D\G/H is compact. When n is even, we find every closed,…
For a homogeneous space $G/H$ of reductive type, we consider the tangential homogeneous space $G_\theta/H_\theta$. In this paper, we give obstructions to the existence of compact Clifford-Klein forms for such tangential symmetric spaces and…
We use a computer-aided approach to prove that there are no standard compact Clifford-Klein forms of homogeneous spaces of exceptional Lie groups. This yields further support for Kobayashi's conjecture about possible compact Clifford-Klein…
In this article we prove that under certain assumptions, a reductive homogeneous space G/H does not admit a solvable compact Clifford-Klein form. This generalizes the well known non-existence theorem of Benoist for nilpotent Clifford-Klein…
We give a classification of irreducible four-dimensional symmetric spaces $G/H$ which admit compact Clifford-Klein forms, where $G$ is the transvection group of $G/H$. For this, we develop a method that applies to particular 1-connected…
This article continues a line of research aimed at solving an important problem of T. Kobayashi of the existence of compact Clifford-Klein forms of reductive homogeneous spaces. We contribute to this topic by showing that almost all…
In this paper we develop algorithms of finding homogeneous spaces of semisimple non-compact Lie groups which do not admit compact Clifford-Klein forms. We propose a computer program which checks if the given homogeneous space has a…
We describe smooth compactifications of certain families of reductive homogeneous spaces such as group manifolds for classical Lie groups, or pseudo-Riemannian analogues of real hyperbolic spaces and their complex and quaternionic…
Let L be a reductive subgroup of a reductive Lie group G. Let G/H be a homogeneous space of reductive type. We provide a necessary condition for the properness of the action of L on G/H. As an application we give examples of spaces that do…
We provide a necessary condition for the existence of a compact Clifford-Klein form of a given homogeneous space of reductive type. The key to the proof is to combine a result of Kobayashi-Ono with an elementary fact that certain two…
We solve the long standing problem of classification of standard compact Clifford-Klein forms of homogeneous spaces of simple non-compact real Lie groups under the extra assumption that $G$, $H$, $L$ are simple and absolutely simple. Then…
In this paper we give the classification of standard compact Clifford-Klein forms corresponding to triples (g,h,l) such that g = h+l and g is a sum of two absolutely simple ideals. The classification is done using Onishchik's results…
We find relations between real root decompositions of triples of Lie algebras corresponding to standard compact Clifford-Klein forms, under the assumption that these triples are not Lie algebra decompositions in the sense of Onishchik. This…
In this paper, we continue the study of the existence problem of compact Clifford-Klein forms from a cohomological point of view, which was initiated by Kobayashi-Ono and extended by Benoist-Labourie and the author. We give an obstruction…
This article studies the volume of compact quotients of reductive homogeneous spaces. Let $G/H$ be a reductive homogeneous space and $\Gamma$ a discrete subgroup of $G$ acting properly discontinuously and cocompactly on $G/H$. We prove that…
Kobayashi [Duke Math. J. (1992)] gave a necessary condition for the existence of compact Clifford-Klein forms in terms of Cartan projections and non-compact dimensions of reductive subgroups. We extend his method to non-reductive subgroups,…
We survey results on compact Clifford-Klein forms of homogeneous spaces, with a focus on recent contributions and organized around approaches via topology, geometry and dynamics. In addition, we survey results on moduli spaces of compact…
Suppose that a finite group $G$ admits a Frobenius group of automorphisms $FH$ with kernel $F$ and complement $H$ such that the fixed-point subgroup of $F$ is trivial: $C_G(F)=1$. In this situation various properties of $G$ are shown to be…
This article discusses the existence problem of a compact quotient of a symmetric space by a properly discontinuous group with emphasis on the non-Riemannian case. Discontinuous groups are not always abundant in a homogeneous space $G/H$ if…