Gradings on Lie algebras with applications to infra-nilmanifolds

Deré, Jonas

doi:10.4171/ggd/390

Cited by 9 publications

(13 citation statements)

References 14 publications

(17 reference statements)

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“…Proof. In [Der,§4], J. Deré constructs a finite-dimensional real nilpotent Lie algebra as a suitable quotient of the free 6-nilpotent real Lie algebra on 4 generators.…”

Section: Existence Of An Invertible Grading Is Not Invariant Under Tamentioning

confidence: 99%

Gradings on Lie algebras, systolic growth, and cohopfian properties of nilpotent groups

Cornulier¹

2016

Bul. Soc. Math. France

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We study the existence of various types of gradings on Lie algebras, such as Carnot gradings or gradings in positive integers, and prove that the existence of such gradings is invariant under extensions of scalars.As an application, we prove that if Γ is a finitely generated nilpotent group, its systolic growth is asymptotically equivalent to its word growth if and only if the Malcev completion of Γ is Carnot.We also characterize when Γ is non-cohopfian, in terms of the existence of a non-trivial grading in non-negative integers, and deduce that this property only depends on its real (or even complex) Malcev completion. n≥0 φ n (Γ) = {1}; such φ is called a dis-cohopf endomorphism. It appears, for a nontrivial group, as a strong negation of being cohopfian. Contents

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“…Proof. In [Der,§4], J. Deré constructs a finite-dimensional real nilpotent Lie algebra as a suitable quotient of the free 6-nilpotent real Lie algebra on 4 generators.…”

Section: Existence Of An Invertible Grading Is Not Invariant Under Tamentioning

confidence: 99%

Gradings on Lie algebras, systolic growth, and cohopfian properties of nilpotent groups

Cornulier¹

2016

Bul. Soc. Math. France

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“…The existence of a positive grading on N Q is equivalent to the existence of a positive grading on the real Lie algebra corresponding to N R , see [5,Theorem 1.4. ] or [8,Theorem 1.2.]. As an immediate consequence of Theorem 4.1, we have the following result.…”

Section: Conjugate Subgroupmentioning

confidence: 68%

“…Proof. Since ρ(F ) is a finite subgroup of Aut(N Q ), there exists a positive grading which is preserved by ρ(F ), see [8,Theorem 2.2.]. In [7,Corollary 3.3.…”

Section: Conjugate Subgroupmentioning

confidence: 99%

Strongly scale-invariant virtually polycyclic groups

Deré¹

2021

Preprint

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A finitely generated group Γ is called strongly scale-invariant if there exists an injective endomorphism ϕ : Γ → Γ with the image ϕ(Γ) of finite index in Γ and the subgroup n>0 ϕ n (Γ) finite. The only known examples of such groups are virtually nilpotent, or equivalently, all examples have polynomial growth. A question by Nekrashevych and Pete asks whether these groups are the only possibilities for such endomorphisms, motivated by the positive answer due to Gromov in the special case of expanding group morphisms.In this paper, we study this question for the class of virtually polycyclic groups, i.e. the virtually solvable groups for which every subgroup is finitely generated. Using the Q-algebraic hull, which allows us to extend the injective endomorphisms of certain virtually polycyclic groups to a linear algebraic group, we show that the existence of such an endomorphism implies that the group is virtually nilpotent. Moreover, we fully characterize which virtually nilpotent groups have a morphism satisfying the condition above, related to the existence of a positive grading on the corresponding radicable nilpotent group. As another application of the methods, we generalize a result of Fel'shtyn and Lee about which maps on infra-solvmanifolds can have finite Reidemeister number for all iterates.

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“…Not every infra-nilmanifold admits expanding maps though, with the first classical example due to [7]. The question which infra-nilmanifolds do admit an expanding map was answered by K. Dekimpe and the first author in [3,5], by showing that it only depends on the existence of a positive grading on the Lie algebra corresponding to the covering nilpotent Lie group N . For the manifolds admitting an Anosov diffeomorphism on the other hand, a lot less is known.…”

Section: Introductionmentioning

confidence: 99%

“…Also describing the infra-nilmanifolds admitting an Anosov diffeomorphism is a challenging question, on which many people have worked over the past decades. Note that the first example of a nilmanifold with an Anosov diffeomorphism that does not admit an expanding map was given in [5]. Unfortunately this example is not explicit and its dimension is unknown.…”

Section: Introductionmentioning

confidence: 99%

On closed manifolds admitting an Anosov diffeomorphism but no expanding map

Deré¹,

Thomas²

2021

Preprint

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A few years ago, the first example of a closed manifold admitting an Anosov diffeomorphism but no expanding map was given. Unfortunately, this example is not explicit and is high-dimensional, although its exact dimension is unknown due to the type of construction. In this paper, we present a family of concrete 12-dimensional nilmanifolds with an Anosov diffeomorphism but no expanding map, where nilmanifolds are defined as the quotient of a 1-connected nilpotent Lie group by a cocompact lattice. We show that this family has the smallest possible dimension in the class of infra-nilmanifolds, which is conjectured to be the only type of manifolds admitting Anosov diffeomorphisms up to homeomorphism. The proof shows how to construct positive gradings from the eigenvalues of the Anosov diffeomorphism under some additional assumptions related to the rank, using the action of the Galois group on these algebraic units.

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Gradings on Lie algebras with applications to infra-nilmanifolds

Cited by 9 publications

References 14 publications

Gradings on Lie algebras, systolic growth, and cohopfian properties of nilpotent groups

Gradings on Lie algebras, systolic growth, and cohopfian properties of nilpotent groups

Strongly scale-invariant virtually polycyclic groups

On closed manifolds admitting an Anosov diffeomorphism but no expanding map

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