Study on solvable Lie groups with specific Weyl connections.
problem Characterizing solvable Lie groups with invariant stretched non-positive Weyl connections.
method Analyzing structure and classification of solvable Lie groups.
result Classification of solvable Lie groups and compact solvmanifolds with invariant SNP connections.
Study finds index of symmetry for solvable 3D Lie groups with left-invariant metrics.
problem Determining the index of symmetry for solvable 3D Lie groups with left-invariant metrics.
method Examined all solvable three-dimensional Lie groups, combined with previous work on unimodular groups.
result Index of symmetry is positive for every solvable 3D Lie algebra with a left-invariant metric, and is never 2.
New metrics with special curvature properties are shown to be parallel in certain Lie groups.
problem Characterizing metrics with harmonic curvature in Lie groups.
method Analyzing left invariant metrics on solvable and low-dimensional Lie groups.
result Left invariant metrics with harmonic curvature are Ricci-parallel in solvable Lie groups and Lie groups of dimension ≤6.
Extended logarithm for solvable elements in mapping class groups.
problem Logarithm of Johnson map extension to solvable elements.
method Extension to exponential solvable elements in mapping class groups using solvable Lie groups.
result Solvability of extended logarithm.
We exhibit a family of infinite, finitely-presented, nilpotent-by-abelian groups. Each member of this family is a solvable S-arithmetic group that is related to Baumslag-Solitar groups, and everyone of these groups has a quasi-isometry group that is virtually a product of a solvable real Lie group and a solvable p-adic…
Paper finds non-positive Weyl connections on Lie groups, confirming a conjecture.
problem Finding non-positive invariant Weyl connections on Lie groups.
method Investigation of completely solvable Lie groups, focusing on SOL group.
result Only SOL admits non-positive Weyl connections, confirming a conjecture.
This paper studies actions of solvable Lie groups on nilpotent Lie groups.
problem Characterizing which solvable Lie groups can act simply transitively on nilpotent Lie groups.
method Using Lie algebra properties and semisimple splitting, the paper provides methods to check for such actions.
result A full description of possibilities for actions up to dimension 4.
Counterexample found for Stein property of certain solvable Lie groups.
problem Stein property of simply connected unimodular solvable Lie groups with left-invariant complex structures.
method Constructing a solvable Lie group with specific properties.
result A simply connected solvable Lie group with a left-invariant complex structure whose universal cover is not Stein.
Proves A-theoretic Farrell-Jones conjecture for virtually solvable groups.
problem Proving the A-theoretic Farrell-Jones conjecture for a specific class of groups.
method Analyzes virtually solvable groups to prove the conjecture.
result Establishes the conjecture for virtually solvable groups and related groups.
The paper classifies isometries on specific Lie groups.
problem Classifying isometries on nonnilpotent, solvable 3D Lie groups.
method Proving automorphisms are the only isometries for rank two Almost-Riemannian Structures.
result A classification result for rank two ARSs on nonnilpotent, solvable 3D Lie groups.
Study finds closed G2-structures on non-solvable Lie groups.
problem Existence of closed G2-structures on non-solvable Lie groups.
method Investigation of left-invariant closed G2-structures on specific Lie groups.
result First examples of closed G2-structures on non-solvable Lie groups.
Classifies Ricci soliton subgroups in specific Lie groups.
problem Classifying Ricci soliton subgroups in solvable Lie groups.
method Analyzing codimension one subgroups of solvable Iwasawa groups and related spaces.
result Classifications of Ricci soliton subgroups in various Lie groups.
It is shown that a closed solvable subgroup of a connected Lie group is compactly generated. In particular, every discrete solvable subgroup of a connected Lie group is finitely generated. Generalizations to locally compact groups are discussed as far as they carry.
We study the subelliptic heat kernels of the CR three dimensional solvable Lie groups. We first classify all left-invariant sub-Riemannian structures on three dimensional solvable Lie groups and obtain representations of these groups. We give expressions for the heat kernels on these groups and obtain heat semigroup gr…
Study ruled minimal surfaces in solvable Lie groups, classifying and constructing examples.
problem Classifying and constructing ruled minimal surfaces in solvable Lie groups.
method Classification and construction of ruled surfaces, parametrization for specific cases.
result Classification and construction of ruled minimal surfaces in solvable Lie groups.
Classifies Lie algebras actions on 3D spaces, focusing on solvable groups.
problem Classifying transitive actions of Lie algebras on 3D spaces.
method Local equivalence and structure of one-dimensional invariant foliations.
result Cannot extend classification to solvable case.
Study on Lie groups with negative Ricci curvature, including open questions and a new cone.
problem Understanding Lie groups with negative Ricci curvature metrics.
method Overview and introduction of a new cone C(n) for solvable Lie algebras.
result Introduction of a new open cone C(n) that parametrizes solvable Lie algebras with negative Ricci curvature metrics.
Let ωg be a Lie algebra valued differential 1-form on a manifold M satisfying the structure equations dωg+21ωg∧ωg=0 where g is solvable. We show that the problem of finding a smooth map ρ:M→G, where G is an n-dimensional so…
Classifies two-step solvable Lie groups with SKT structures.
problem Classifying Lie groups with SKT structures.
method Shear construction and analysis of SKT shear data on Abelian Lie algebras.
result Large part of the classification for two-step solvable SKT algebras of dimension six.
We show that the Cheeger isoperimetric constant of a solvable simply connected Lie group G with Lie algebra $\G$ is $h(G)=\max_{H\in\G,||H||=1} \tr(\ad (H))$.
In this paper we introduce two new methods for constructing harmonic morphisms from solvable Lie groups. The first method yields global solutions from any simply connected nilpotent Lie group and from any Riemannian symmetric space of non-compact type and rank r≥3. The second method provides us with global solutio…
Let G be a simply connected, solvable Lie group and Γ a lattice in G. The deformation space D(Γ,G) is the orbit space associated to the action of $\Aut(G)$ on the space X(Γ,G) of all lattice embeddings of Γ into G. Our main result generalises the classical rigidity theorems of Mal'tsev…
The paper proves a theorem about completely solvable Lie foliations on compact manifolds.
problem Generalizing Haefliger's theorem to completely solvable Lie foliations.
method Proves that every completely solvable Lie foliation on a compact manifold is the inverse image of a homogeneous foliation.
result Every completely solvable Lie foliation on a compact manifold is the inverse image of a homogeneous foliation.
Paper generalizes results from nilpotent Lie algebras to broader types.
problem Generalizing results from nilpotent Lie algebras to broader types.
method Examined Einstein Lorentzian unimodular and solvable Lie algebras.
result Key results from nilpotent Lie algebras still hold in broader settings.
Study explores solvable Lie groups' actions on closed Lorentzian manifolds.
problem Understanding conformal actions of solvable Lie groups on closed Lorentzian manifolds.
method Analyzes the identity component of the conformal group and uses algebraic hypotheses.
result Establishes conditions for conformal flatness and local embeddings.
The paper classifies Lie groups with specific quasi-Einstein metrics.
problem Investigating Lie groups with quasi-Einstein metrics.
method Complete classification of nilpotent and unimodular solvable Lie groups.
result Classification of Lie groups admitting quasi-Einstein metrics.
The paper extends Heintze-Kobayashi-Wolf theory to negatively curved homogeneous Finsler manifolds.
problem Understanding negatively curved homogeneous Finsler manifolds.
method Generalizing Heintze-Kobayashi-Wolf theory to homogeneous Finsler geometry, proving two main theorems.
result Negatively curved homogeneous Finsler manifolds are isometric to Lie groups with specific properties.
The paper constructs Einstein Sasaki metrics on solvable Lie groups.
problem Constructing left-invariant Einstein pseudo-Riemannian Sasaki metrics on solvable Lie groups.
method Characterizing pseudo-Kähler structures and derivations giving rise to Sasaki-Einstein metrics.
result Classification of z-standard Sasaki solvable Lie algebras of dimension ≤7. In this paper we describe the geodesics of a left-invariant sub-Riemannian metric on the three-dimensional solvable Lie group SOLV−.
New examples of Laplacian solitons found on solvable Lie groups.
problem Finding closed Laplacian solitons with finite-time singularities.
method Left-invariant G2-structures on solvable Lie groups.
result First examples of closed Laplacian solitons with finite-time singularities.
Flat Hermitian Lie algebras are always Kähler.
problem Classifying Lie groups with Hermitian structures that are flat.
method Analysis on the Hermitian geometry of 2-step solvable Lie groups.
result Flat Hermitian Lie algebras are Kähler.
We consider the question of whether a given solvable Lie group admits a left-invariant metric of strictly negative Ricci curvature. We give necessary and sufficient conditions of the existence of such a metric for the Lie groups the nilradical of whose Lie algebra is either abelian or Heisenberg or standard filiform, a…
Characterizes connections on normal distributions manifold.
problem Geometric characterization of connections on normal distributions.
method Homogeneous statistical manifold structure and Lie group analysis.
result Geometric characterization of α-connections on Lie group. A strong KT (SKT) manifold consists of a Hermitian structure whose torsion three-form is closed. We classify the invariant SKT structures on four-dimensional solvable Lie groups. The classification includes solutions on groups that do not admit compact four-dimensional quotients. It also shows that there are solvable g…
Study confirms optimal bounds for group cohomology of Lie groups.
problem Optimal bounds for group cohomology of Lie groups.
method Combining complementary vanishings with spectral sequences and quasi-isometry invariance.
result Non-vanishing of group Lp-cohomology for large p and equal degree to rank. The study explores maximal symmetry in Ricci solitons on Lie groups.
problem Maximal symmetry in left-invariant Riemannian metrics and Ricci solitons.
method Analysis of left-invariant metrics and Ricci solitons on Lie groups, using tools from previous work on Einstein metrics.
result Expanding homogeneous Ricci solitons have maximal isometry algebras but not always maximal isometry groups.
Study Weyl-Einstein structures on conformal solvmanifolds, proving Einstein property and classifying metrics.
problem Characterize Weyl-Einstein structures on conformal solvmanifolds.
method Analyzing left-invariant metrics and using conformal Lie group structures.
result Every conformal solvmanifold with Weyl-Einstein structure is Einstein.
In this work we investigate solvable and nilpotent Lie groups with special metrics. The metrics of interest are left-invariant Einstein and algebraic Ricci soliton metrics. Our main result shows that the existence of a such a metric is intrinsic to the underlying Lie algebra. More precisely, we show how one may determi…
This note classifies splittable lattices in a specific Lie group.
problem Classifying splittable lattices in a metabelian solvable Lie group.
method Description and classification of splittable lattices in G:=RntimesηRm. result Classification of splittable lattices in the specified Lie group.
In this paper, we shall use a method based on the theory of extensions of left-symmetric algebras to classify complete left-invariant affine real structures on solvable non-unimodular three-dimensional Lie groups.
New Lie algebras from quivers lead to rigid Ricci solitons.
problem Constructing Lie algebras from quivers to study geometric structures.
method Using finite quivers without cycles to construct solvable Lie algebras and proving their geometric properties.
result Simply-connected Lie groups corresponding to these Lie algebras admit left-invariant Ricci solitons, and when quivers are oriented multi-trees, these groups are rigid.
New solitons found for G2-Laplacian flow on Lie groups.
problem Existence of solitons for G2-Laplacian flow. method Existence proof for expanding and steady solitons.
result First example of non-extremally Ricci pinched steady soliton.
By using results by D. Witte on the superigidity of lattices in solvable Lie groups we get a different proof of a recent remarkable result obtained by D. Guan on the de Rham cohomology of a compact solvmanifold, i.e. of a quotient of a connected and simply connected solvable Lie group G by a lattice Γ. This result …
Study LCP structures on solvmanifolds, complete list up to 5 dimensions.
problem Characterize LCP structures on solvable Lie groups.
method Classify LCP structures on Lie groups, focus on solvable unimodular case.
result Complete list of solvable unimodular Lie algebras up to dimension 5 with LCP structures.
This work builds on the foundation laid by Gordon and Wilson in the study of isometry groups of solvmanifolds, i.e. Riemannian manifolds admitting a transitive solvable group of isometries. We restrict ourselves to a natural class of solvable Lie groups called almost completely solvable; this class includes the complet…
The paper splits local rigidity into vertical and horizontal types.
problem Local rigidity of Clifford-Klein forms in homogeneous spaces.
method Introducing a splitting of local rigidity into vertical and horizontal rigidity.
result Refined results and a new approach to Baklouti's conjecture.
Minimal lattice generating set size linked to group co-volume.
problem Determining minimal generating set size for lattices in Lie groups.
method Relies on semi-simple and solvable cases, extends to amenable groups.
result Bounding minimal generating set size by co-volume of projection.
We answer in the affirmative a question posed by Ivanov and Vassilev on the existence of a seven dimensional quaternionic contact manifold with closed fundamental 4-form and non-vanishing torsion endomorphism. Moreover, we show an approach to the classification of seven dimensional solvable Lie groups having an integra…