Searcharxiv⌕ Search

arXiv subjects

Maciej Bochenski

Publications and source records attributed to Maciej Bochenski.

At least 19 recordsLinked to original sources
arXiv2025

Exotic proper actions on homogeneous spaces via convex cocompact representations

We construct a series of homogeneous spaces G/H of reductive type which admit proper actions of discrete subgroups of G isomorphic to cocompact lattices of O(n,1) (n=2,3,4) but do not admit proper actions of non-compact semisimple subgroups of G. The existence of such homogeneous spaces was previously not known even for n=2. Our construction of proper actions of discrete subgroups is based on Guéritaud-Kassel's work on convex cocompact subgroups of O(n,1) and Danciger-Guéritaud-Kassel's work on right-angled Coxeter groups. On the other hand, the non-existence of proper actions of non-compact semisimple subgroups is proved by the theory of nilpotent orbits and elementary combinatorics.

math.GR↗
arXiv2025

A solution of the problem of standard compact Clifford-Klein forms

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 the result is that standard compact Clifford-Klein forms always arise from triples $(\mathfrak{g},\mathfrak{h},\mathfrak{l})$ of real Lie algebras such that $\mathfrak{h}\subset\mathfrak{g},\mathfrak{l}\subset\mathfrak{g}$, $\mathfrak{g}$ is simple and absolutely simple, $\mathfrak{h},\mathfrak{l}$ are (non-compact) reductive, $\mathfrak{g}=\mathfrak{h}+\mathfrak{l}$, and the intersection $\mathfrak{h}\cap\mathfrak{l}$ is compact. The consequence of this is the following characterization of proper co-compact actions of reductive Lie subgroups $L\subset G$ on a homogeneous spaces $G/H$ determined by absolutely simple real Lie group $G$ and a closed reductive subgroup $H$: $L$ acts on $G/H$ properly and co-compactly if and only if $G=H\cdot L$ and $H\cap L$ is compact.

math.DG↗
arXiv2022

Standard compact Clifford-Klein forms and Lie algebra decompositions

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 enables us to find new classes of homogeneous spaces of simple real Lie groups which do not admit standard compact Clifford-Klein forms. In particular, we show that proper R-regular subalgebras of simple real Lie algebras never generate homogeneous spaces which admit compact standard Cliffrod-Klein forms.

math.RT↗
arXiv2022

Stretched non-positive Weyl connections on solvable Lie groups

We determine the structure of solvable Lie groups endowed with invariant stretched non-positive Weyl connections and find classes of solvable Lie groups admitting and not admitting such connections. In dimension 4 we fully classify solvable Lie groups and compact solvmanifolds which admit invariant SNP connections.

math.DG↗
arXiv2021

Homogeneous spaces of real simple Lie groups with proper actions of non virtually abelian discrete subgroups: a calculational approach

Let G be a simple non-compact linear connected Lie group and H be a closed non-compact semisimple subgroup. We are interested in finding classes of homogeneous spaces G/H admitting proper actions of discrete non virtually abelian subgroups of G. We develop an algorithm for finding such homogeneous spaces. As a testing example we obtain a list of all non-compact homogeneous spaces G/H admitting proper action of a discrete and non virtually abelian subgroup of G in the case when G has rank at most 8, and H is a maximal proper semisimple subgroup.

math.GR↗
arXiv2020

On locally homogeneous pseudo-Riemannian compact einstein manifolds

We ask a general question: what are locally homogeneous compact pseudo-Riemannian Einstein manifolds? We show that any standard compact Clifford-Klein form of a simple non-compact Lie group admits at least one Einstein metric. We conjecture that these are basically the only possible locally homogeneous Einstein pseudo-Riemannian compact manifolds using T. Kobayashi's conjecture as a guiding principle.

math.DG↗
arXiv2019

On the Hirzebruch-Kobayashi-Ono proportionality principle and the non-existence of compact solvable Clifford-Klein forms of certain homogeneous spaces

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 symmetric spaces and 3-symmetric spaces do not admit solvable compact CliffordfKlein forms (with several possible exceptions). Our basic tool is a combination of the Hirzebruch-Kobayashi-Ono proportionality principle with the theory of syndetic hulls. Using this, we prove a general theorem which yields a sufficient condition for the non-existence of compact solvable CliffordKlein forms.

math.DG↗
arXiv2019

On solvable compact Clifford-Klein forms

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 forms. This generalization works for a particular class of homogeneous spaces determined by "very regular" embeddings of H into G.

math.DG↗
arXiv2019

Non-existence of standard compact Clifford-Klein forms of homogeneous spaces of exceptional Lie groups

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 forms. On one hand, our approach is based on the algorithms developed in this work which eliminate the majority of possibilities. On the other hand, we complete the proof using the algorithmic methods of classifying semisimple subalgebras in simple real Lie algebras developed by Faccin and de Graaf, as well as by calculating invariants like a-hyperbolic rank.

math.DG↗
arXiv2016

Proper SL(2,R)-actions on homogeneous spaces

We study the existence problem of proper actions of SL(2,R) on homogeneous spaces G/H of reductive type. Based on Kobayashi's properness criterion [Math. Ann. (1989)], we show that G/H admits a proper SL(2,R)-action via G if a maximally split abelian subspace of Lie H is included in the wall defined by a restricted root of Lie G. We also give a number of examples of such G/H.

math.GR↗
arXiv2016

On a classification of fat bundles over compact homogeneous spaces

This article deals with fat bundles. Berard-Bergery classified all homogeneous bundles of that type. We ask a question of a possibility to generalize his description in the case of arbitrary G-structures over homogeneous spaces. We obtain necessary conditions for the existence of such bundles. These conditions yield a kind of classification of fat bundles associated with G-structures over compact homogeneous spaces provided that the connection in a G-structure is canonical.

math.DG↗
arXiv2015

A restriction on proper actions on homogeneous spaces of reductive type

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 not admit standard compact Clifford-Klein forms.

math.GR↗
arXiv2015

On symplectically fat twistor bundles

This paper deals with the question when are twistor bundles over homogeneous spaces symplectically fat? It shows that twistor bundles over even dimensional Grassmannians of maximal rank have this property.

math-ph↗
arXiv2015

New constructions of symplectically fat fiber bundles

This work is devoted to new constructions of symplectically fat fiber bundles. The latter are constructed in two ways: using the Kirwan map and expressing the fatness condition in terms of the isotropy representation related to the G-structure over some homogeneous spaces.

math.DG↗
arXiv2015

Proper actions on strongly regular homogeneous spaces

Let G/H be a strongly regular homogeneous space such that H is a Lie group of inner type. We show that G/H admits a proper action of a discrete non-virtually abelian subgroup of G if and only if G/H admits a proper action of a subgroup L of G locally isomorphic to SL(2,R). We classify all such spaces.

math.DG↗
arXiv2014

Clifford-Klein forms and a-hyperbolic rank

The porpose of this article is to introduce and investigate properties of a tool (the a-hyperbolic rank) which enables us to obtain new examples of homogeneous spaces G/H which admit and do not admit almost compact Clifford-Klein forms. We achieve this goal by exploring in greater detail the technique of adjoint orbits developed by Okuda combined with the well-known conditions of Benoist. We find easy-to-check conditions on G and H expressed directly in terms of the Satake diagrams of the corresponding Lie algebras, in cases when G is a real form of a complex Lie group of types A, D or E6. One of the advantages of this approach is the fact that we don't need to know the embedding of H into G. Using the a-hyperbolic rank we also show that the homogeneous space G/H of reductive type of the real form of E6 (of the IV type) admits compact Clifford-Klein forms if and only if H is compact. Inspired by the work of Okuda on symmetric spaces G/H we classify all 3-symmetric spaces admitting almost compact Clifford-Klein forms.

math.GR↗