arXiv subjects

Jonas Deré

Publications and source records attributed to Jonas Deré.

30 records · Page 2Linked to original sources

arXiv2020

Orthogonal bi-invariant complex structures on metric Lie algebras

This paper studies how many orthogonal bi-invariant complex structures exist on a metric Lie algebra over the real numbers. Recently, it was shown that irreducible Lie algebras which are additionally $2$-step nilpotent admit at most one orthogonal bi-invariant complex structure up to sign. The main result generalizes this statement to metric Lie algebras with any number of irreducible factors and which are not necessarily $2$-step nilpotent. It states that there are either $0$ or $2^k$ such complex structures, with $k$ the number of irreducible factors of the metric Lie algebra. The motivation for this problem comes from differential geometry, for instance to construct non-parallel Killing-Yano $2$-forms on nilmanifolds or to describe the compact Chern-flat quasi-Kähler manifolds. The main tool we develop is the unique orthogonal decomposition into irreducible factors for metric Lie algebras with no non-trivial abelian factor. This is a generalization of a recent result which only deals with nilpotent Lie algebras over the real numbers. Not only do we apply this fact to describe the orthogonal bi-invariant complex structures on a given metric Lie algebra, but it also gives us a method to study different inner products on a given Lie algebra, computing the number of irreducible factors and orthogonal bi-invariant complex structures for varying inner products.

math.DG↗

arXiv2020

Simply transitive NIL-affine actions of solvable Lie groups

Every simply connected and connected solvable Lie group $G$ admits a simply transitive action on a nilpotent Lie group $H$ via affine transformations. Although the existence is guaranteed, not much is known about which Lie groups $G$ can act simply transitive on which Lie groups $H$. So far the focus was mainly on the case where $G$ is also nilpotent, leading to a characterization depending only on the corresponding Lie algebras and related to the notion of post-Lie algebra structures. This paper studies two different aspects of this problem. First, we give a method to check whether a given action $ρ: G \to \operatorname{Aff}(H)$ is simply transitive by looking only at the induced morphism $φ: \mathfrak{g} \to \operatorname{aff}(\mathfrak{h})$ between the corresponding Lie algebras. Secondly, we show how to check whether a given solvable Lie group $G$ acts simply transitive on a given nilpotent Lie group $H$, again by studying properties of the corresponding Lie algebras. The main tool for both methods is the semisimple splitting of a solvable Lie algebra and its relation to the algebraic hull, which we also define on the level of Lie algebras. As an application, we give a full description of the possibilities for simply transitive actions up to dimension $4$.

math.GR↗

arXiv2019

A note on the structure of underlying Lie algebras

Every Lie algebra over a field $E$ gives rise to new Lie algebras over any subfield $F \subseteq E$ by restricting the scalar multiplication. This paper studies the structure of these underlying Lie algebra in relation to the structure of the original Lie algebra, in particular the question how much of the original Lie algebra can be recovered from its underlying Lie algebra over subfields $F$. By introducing the conjugate of a Lie algebra we show that in some specific cases the Lie algebra is completely determined by its underlying Lie algebra. Furthermore we construct examples showing that these assumptions are necessary. As an application, we give for every positive $n$ an example of a real $2$-step nilpotent Lie algebra which has exactly $n$ different bi-invariant complex structures. This answers an open question by Di Scala, Lauret and Vezzoni motivated by their work on quasi-Kähler Chern-flat manifolds in differential geometry. The methods we develop work for general Lie algebras and for general Galois extensions $F \subseteq E$, in contrast to the original question which only considered nilpotent Lie algebras of nilpotency class $2$ and the field extension $\mathbb{R} \subseteq \mathbb{C}$. We demonstrate this increased generality by characterizing the complex Lie algebras of dimension $\leq 4$ which are defined over $\mathbb{R}$ and over $\mathbb{Q}$.

math.RA↗

arXiv2018

NIL-affine crystallographic actions of virtually polycyclic groups

A classical result by K.B. Lee states that every group morphism between almost crystallographic groups is induced by an affine map on the nilpotent Lie group whereon these groups by definition act. It is the main technique for studying morphisms between virtually nilpotent groups, having important applications in fixed point theory, for example as a tool to compute Nielsen numbers for self-maps on infra-nilmanifolds. In this paper we generalize this result to morphisms between virtually polycyclic groups, which are known to act crystallographically by affine transformations on a nilpotent Lie group. The main method is to study the special class of translation-like actions, which behave well under taking subgroups and restricting the action to invariant right cosets.

math.GR↗

arXiv2018

Effective Subgroup Separability of Finitely Generated Nilpotent Groups

This paper studies effective separability for subgroups of finitely generated nilpotent groups and more broadly effective subgroup separability of finitely generated nilpotent groups. We provide upper and lower bounds that are polynomial with respect to the logarithm of the word length for infinite index subgroups of nilpotent groups. In the case of normal subgroups, we provide an exact computation generalizing work of the second author. We introduce a function that quantifies subgroup separability, and we provide polynomial upper and lower bounds. We finish by demonstrating that our results extend to virtually nilpotent groups.

math.GR↗

arXiv2018

Effective Twisted Conjugacy Separability of Nilpotent Groups

This paper initiates the study of effective twisted conjugacy separability for finitely generated groups, which measures the complexity of separating distinct twisted conjugacy classes via finite quotients. The focus is on nilpotent groups, and our main result shows that there is a polynomial upper bound for twisted conjugacy separability. That allows us to study regular conjugacy separability in the case of virtually nilpotent groups, where we compute a polynomial upper bound as well. As another application, we improve the work of the second author by giving a precise calculation of conjugacy separability for finitely generated nilpotent groups of nilpotency class 2.

math.GR↗

arXiv2017

On Ricci negative solvmanifolds and their nilradicals

In the homogeneous case, the only curvature behavior which is still far from being understood is Ricci negative. In this paper, we study which nilpotent Lie algebras admit a Ricci negative solvable extension. Different unexpected behaviors were found. On the other hand, given a nilpotent Lie algebra, we consider the space of all the derivations such that the corresponding solvable extension has a metric with negative Ricci curvature. Using the nice convexity properties of the moment map for the variety of nilpotent Lie algebras, we obtain a useful characterization of such derivations and some applications.

math.DG↗

arXiv2016

Gradings on Lie algebras with applications to infra-nilmanifolds

In this paper, we study positive as well as non-negative and non-trivial gradings on finite dimensional Lie algebras. We give a different proof that the existence of such a grading on a Lie algebra is invariant under taking field extensions, a result very recently obtained by Y. Cornulier. Similarly, we prove that given a grading of one these types and a finite group of automorphisms, there always exist a positive grading which is preserved by this group. From these results we conclude that the existence of an expanding map or a non-trivial self-cover on an infra-nilmanifold depends only on the covering Lie group. Another application is the construction of a nilmanifold admitting an Anosov diffeomorphism but no non-trivial self-covers and in particular no expanding maps, which is the first known example of this type.

math.RA↗

arXiv2015

Periodic and eventually periodic points of affine infra-nilmanifold endomorphisms

In this paper, we study the periodic and eventually periodic points of affine infra-nilmanifold endomorphisms. On the one hand, we give a sufficient condition for a point of the infra-nilmanifold to be (eventually) periodic. In this way we show that if an affine infra-nilmanifold endomorphism has a periodic point, then its set of periodic points forms a dense subset of the manifold. On the other hand, we deduce a necessary condition for eventually periodic points from which a full description of the set of eventually periodic points follows for an arbitrary affine infra-nilmanifold endomorphism.

math.AT↗

arXiv2015

A new method for constructing Anosov Lie algebras

It is conjectured that every manifold admitting an Anosov diffeomorphism is, up to homeomorphism, finitely covered by a nilmanifold. Motivated by this conjecture, an important problem is to determine which nilmanifolds admit an Anosov diffeomorphism. The main theorem of this article gives a general method for constructing Anosov diffeomorphisms on nilmanifolds. As a consequence, we give counterexamples to a corollary of the classification of low-dimensional nilmanifolds with Anosov diffeomorphisms and a correction to this statement is proven. This method also answers some open questions about the existence of Anosov diffeomorphisms which are minimal in some sense.

math.DS↗

arXiv2014

Expanding maps and non-trivial self-covers on infra-nilmanifolds

Every expanding map on a closed manifold is topologically conjugate to an expanding map on an infra-nilmanifold, but not every infra-nilmanifold admits an expanding map. In this article we give a complete algebraic characterization of the infra-nilmanifolds admitting an expanding map. We show that, just as in the case of Anosov diffeomorphisms, the existence of an expanding map depends only on the rational holonomy representation of the infra-nilmanifold. A similar characterization is also given for the infra-nilmanifolds with a non-trivial self-cover, which corresponds to determining which almost-Bieberbach groups are co-Hopfian. These results provide many new examples of infra-nilmanifolds without non-trivial self-covers or expanding maps.

math.DS↗

arXiv2014

Existence of Anosov diffeomorphisms on infra-nilmanifolds modeled on free nilpotent Lie groups

An infra-nilmanifold is a manifold which is constructed as a quotient space $Γ\backslash G$ of a simply connected nilpotent Lie group $G$, where $Γ$ is a discrete group acting properly discontinuously and cocompactly on $G$ via so called affine maps. The manifold $Γ\backslash G$ is said to be modeled on the Lie group $G$. This class of manifolds is conjectured to be the only class of closed manifolds allowing an Anosov diffeomorphism. However, it is far from obvious which of these infra--nilmanifolds actually do admit an Anosov diffeomorphism. In this paper we completely solve this question for infra-nilmanifolds modeled on a free $c$--step nilpotent Lie group.

math.DS↗