On quasi-isometry invariants associated to a Heintze group

Piaggio, Matí­as; Sequeira, Emiliano

doi:10.1007/s10711-016-0214-9

Cited by 10 publications

(18 citation statements)

References 15 publications

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“…The plain horizontal lines denote the separations between O(log)-bilipschitz equivalence classes that can be deduced from [Pal20] and Theorem E. The dash line remains unknown when µ = 1+λ. The isomorphism type of N ⋊ α R is generally not determined by N and Jordan(α) alone; see the 6-dimensional example after Theorem 1.3 in [CPS17].…”

Section: (C3) Implies (C2) Since It Acts Geometrically On Gromov-hype...mentioning

confidence: 99%

On the logarithmic coarse structures of Lie groups and hyperbolic spaces

Pallier¹

2021

Preprint

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We characterize the Lie groups with finitely many connected components that are O(u)-bilipschitz equivalent (almost quasiisometric in the sense that the sublinear function u replaces the additive bounds of quasiisometry) to the real hyperbolic space, or to the complex hyperbolic plane. The characterizations are expressed in terms of deformations of Lie algebras and in terms of pinching of sectional curvature of left-invariant Riemannian metrics in the real case. We also compare sublinear bilipschitz equivalence and coarse equivalence, and prove that every coarse equivalence between the logarithmic coarse structures of geodesic spaces is a O(log)-bilipschitz equivalence. The Lie groups characterized are exactly those whose logarithmic coarse structure is equivalent to that of a real hyperbolic space or the complex hyperbolic plane.

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Section: (C3) Implies (C2) Since It Acts Geometrically On Gromov-hype...mentioning

confidence: 99%

On the logarithmic coarse structures of Lie groups and hyperbolic spaces

Pallier¹

2021

Preprint

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“…Sequeira recently recovered it using relative L p -cohomology [Seq19, Theorem 1.5]. The quasiisometry invariance of the characteristic polynomial of α holds in general [CPS17].…”

Section: Quasiisometric Classification Of Diagonalizable Heintze Groupsmentioning

confidence: 99%

Sublinear quasiconformality and the large-scale geometry of Heintze groups

Pallier

2020

Conform. Geom. Dyn.

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This article analyzes sublinearly quasisymmetric homeomorphisms (generalized quasisymmetric mappings), and draws applications to the sublinear large-scale geometry of negatively curved groups and spaces. It is proven that those homeomorphisms lack analytical properties but preserve a conformal dimension and appropriate function spaces, distinguishing certain (nonsymmetric) Riemannian negatively curved homogeneous spaces, and Fuchsian buildings, up to sublinearly biLipschitz equivalence (generalized quasiisometry).

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“…Note that diagonal Heintze groups form a subset of purely real Heintze groups. By [CPS17] the quasiisometric classification problem of diagonal Heintze groups reduces to an algebraic problem of finding all the possible derivations defining non-isomorphic Heintze groups. Proposition 4.7 is a tool for tackling this algebraic problem using positive gradings.…”

Section: Name K Smentioning

confidence: 99%

“…For instance, it is not known whether there exists a non-stratifiable positively gradable Lie group that is quasi-isometric to its asymptotic cone, nor whether all large-scale contractible groups are positively gradable, see [Cor19,Question 7.9]. The quasi-isometric classification is open also for Heintze groups, see [CPS17] for some known results.…”

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confidence: 99%

Gradings for nilpotent Lie algebras

Hakavuori¹,

Kivioja²,

Moisala³

et al. 2020

Preprint

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We present a constructive approach to torsion-free gradings of Lie algebras. Our main result is the computation of a maximal grading. Given a Lie algebra, using its maximal grading we enumerate all of its torsion-free gradings as well as its positive gradings. As applications, we classify gradings in low dimension, we consider the enumeration of Heintze groups, and we give methods to find bounds for non-vanishing ℓ q,p cohomology. Contents 1. Introduction 1.1. Overview 1.2. Main results 1.3. Structure of the paper 2. Gradings 2.1. Gradings and equivalences 2.2. Universal gradings 2.3. Gradings induced by tori 2.4. Maximal gradings 2.5. Enumeration of torsion-free gradings 3. Constructions 3.1. Stratifications 3.2. Positive gradings 3.3. Maximal gradings 4. Applications 4.1. Structure from maximal gradings 4.2. Classification of gradings in low dimension 4.3. Enumerating Heintze groups 4.4. Bounds for non-vanishing ℓ q,p cohomology Appendix A. Existence of a positive realization References

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On quasi-isometry invariants associated to a Heintze group

Cited by 10 publications

References 15 publications

On the logarithmic coarse structures of Lie groups and hyperbolic spaces

On the logarithmic coarse structures of Lie groups and hyperbolic spaces

Sublinear quasiconformality and the large-scale geometry of Heintze groups

Gradings for nilpotent Lie algebras

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