Study on subgroups' evolution in 3D Lie groups using mean curvature flow.
problem Existence of solutions to Mean Curvature Flow for 2D Lie subgroups in 3D Lie groups.
method Investigation of Lie groups with fixed left-invariant metrics, focusing on non-unimodular cases.
result Evolution of Lie subgroups is self-similar for abelian subgroups, but not for others.
The study examines discrete subgroups of Lie groups and their residual finiteness.
problem Determining when discrete subgroups of Lie groups are residually finite.
method Analyzes known results and poses open questions.
result Answers to open questions will provide a comprehensive understanding of residual finiteness in Lie groups.
Example found of subgroup not a lattice in product of Lie groups
problem Finding irreducible discrete subgroups that are not lattices
method Produced an example in SL(2,R)imesSL(2,R) result Example of subgroup not a lattice in product of Lie groups
The paper studies Riemannian metrics on Lie groups with specific commutator subgroups.
problem Investigating Riemannian metrics on Lie groups with commutator subgroups of dimensions 1 and 2.
method Explicitly provided Levi-Civita connection, sectional curvature, and Ricci curvature; computed necessary and sufficient conditions for Ricci solitons; characterized Ricci solitons on Lie groups with one-dimensional commutator subgroups; examined indecomposable Lie groups with two-dimensional commutator subgroups.
result Characterization of all Ricci solitons on Lie groups with one-dimensional commutator subgroups and examination of indecomposable Lie groups with two-dimensional commutator subgroups.
Cohomology of Lie group quotient equals Lie algebra cohomology.
problem Cohomology of Lie group quotients.
method Diffeological de Rham cohomology and Lie algebra cohomology.
result Cohomology of G/H equals Lie algebra cohomology of g/h. Improved homological dimension for certain subgroups in Lie groups.
problem Determining homological dimensions of discrete subgroups in Lie groups.
method Using recent results and properties of injectivity radius, the homological dimension gap is calculated.
result Infinite volume torsion-free subgroups of higher rank Lie groups have a homological dimension gap of at least 1/8 of the real rank.
We discuss a Moser type argument to show when a deformation of a Lie group homomorphism and of a Lie subgroup is trivial. For compact groups we obtain stability results.
The study finds discrete subgroups with full limit sets in higher rank Lie groups.
problem Finding discrete subgroups with full limit sets in higher rank Lie groups.
method Analyzing real semi-simple Lie groups of higher rank and providing criteria for discrete subgroups of G=SL(3,R). result Existence of discrete subgroups with full limit sets in higher rank Lie groups.
Characterizes knotted subgroups of Lie groups and provides examples.
problem Defining and understanding knotted subgroups of Lie groups.
method Geometric equivalence, one-parameter subgroups, infinitesimal elements, canonical forms, spectrum analysis.
result Completely classified knotted subgroups of SL(2,R) and SL(3,R).
The paper finds free semigroups in dense subgroups of Lie groups with critical exponents arbitrarily close to the subgroup's.
problem Finding free semigroups with critical exponents arbitrarily close to a subgroup's in dense subgroups of Lie groups.
method Analyzing Zariski dense discrete subgroups of Lie groups, showing the existence of free semigroups with critical exponents arbitrarily close to the subgroup's.
result The existence of free semigroups with critical exponents arbitrarily close to the subgroup's in dense subgroups of Lie groups.
Study on geometry and dynamics of transverse subgroups.
problem Understanding the geometry and dynamics of transverse subgroups.
method Survey of recent research on semi-simple Lie groups.
result Recent findings on transverse subgroups of semi-simple Lie groups.
This report aims at giving a general overview on the classification of the maximal subgroups of compact Lie groups (not necessarily connected). In the first part, it is shown that these fall naturally into three types: (1) those of trivial type, which are simply defined as inverse images of maximal subgroups of the cor…
Study of Hermitian structures on Lie groups with 2D commutator subgroups.
problem Classifying Hermitian structures on Lie groups with specific commutator subgroups.
method Explicit classification of Type I and Type II structures, computation of Bismut connections, and examples of Kahler structures.
result Classification of Kahler structures within Type I and Type II structures.
The paper constructs Anosov representations for specific types of groups.
problem Constructing Anosov representations for certain groups.
method Analyzing uniform lattices and their extensions, proving existence of Anosov embeddings.
result Examples of one-ended hyperbolic groups admit Anosov embeddings into higher-rank Lie groups.
Characterizes isolated compact subgroups in Lie groups.
problem Identifying isolated compact subgroups in Lie groups.
method Characterization based on intrinsic structure, irrelevant ambient group details.
result Characterization of isolated compact subgroups depends only on intrinsic structure.
This paper classifies Ricci soliton subgroups in a specific type of nilpotent group.
problem Classifying Ricci soliton subgroups in a specific type of nilpotent group.
method Using the properties of nilpotent Iwasawa groups and Lie subgroups.
result Classification of codimension one Lie subgroups of nilpotent Iwasawa groups that are Ricci solitons.
Study answers arithmeticity question for normal subgroup of lattices.
problem Arithmeticity of discrete subgroups of semisimple Lie groups with dense commensurators.
method Examined normal subgroups of lattices in semisimple Lie groups.
result Positive answer to Greenberg-Shalom's question for lattices.
In this article we classify all connected H-irreducible Lie subgroups of Sp(1,n) up to conjugacy.
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.
Study uses Lie group subgroups to identify special subspaces in calibrations.
problem Identifying calibrated subspaces in Lie groups.
method Utilizes the principal three-dimensional subgroup of a simple Lie group.
result Identifies certain special subspaces as calibrated for invariant forms.
Researchers found sub-Lorentzian geodesics on a specific Lie subgroup.
problem Finding geodesics on a specific Lie subgroup with a sub-Lorentzian metric.
method Formulated a time-anti-optimal control problem, applied Pontryagin's minimum principle, and used geodesics and shortest arcs of a sub-Riemannian metric.
result Discovered sub-Lorentzian nonspacelike geodesics and longest arcs.
Study critical exponents in normal subgroups of higher rank Lie groups.
problem Understanding critical exponents in normal subgroups of higher rank Lie groups.
method Analyzing subgroups and their critical exponents in a higher rank semi-simple Lie group.
result Critical exponents of normal subgroups coincide under certain conditions.
The study extends Dehn filling to Lie groups, ensuring geometric properties.
problem Generalizing Dehn filling to semisimple Lie groups.
method Analyzing deformations of subgroups and their geometric properties.
result Extended geometrically finite subgroups can be deformed while maintaining properties.
Reduces field theories on principal bundles by a subgroup, deriving reduced equations.
problem Hamiltonian field theories on principal G-bundles with invariant densities.
method Lie-Poisson reduction using covariant bracket formulation.
result Derives reduced observables, brackets, and equations of motion for field theories.
Survey on Coxeter groups for Lie group examples.
problem Understanding Coxeter groups and their applications.
method Constructing discrete subgroups of Lie groups using Coxeter groups.
result Coxeter groups provide new examples in discrete subgroups of Lie groups.
This paper classifies Ricci solitons in complex hyperbolic spaces.
problem Understanding Ricci solitons in complex hyperbolic spaces.
method Analyzing homogeneous expanding Ricci solitons as submanifolds of complex hyperbolic spaces.
result Classification and analysis of Lie subgroups with Ricci soliton induced metric in complex hyperbolic spaces.
Study Riemann-Finsler geometry on tangent bundles of Lie groups with 2D commutator subgroup.
problem Characterize Riemannian and Finslerian properties of tangent bundles of Lie groups with specific commutator subgroups.
method Investigate sectional curvatures, define Randers metrics, and compute flag curvatures on tangent bundles.
result Explicit formulas for Riemannian curvature tensor on tangent bundles of Lie groups with 2D commutator subgroup.
We are interested in the class, in the Elie Cartan sense, of left invariant forms on a Lie group. We construct the class of Lie algebras provided with a contact form and classify the frobeniusian Lie algebras up to a contraction. We also study forms which are invariant by a subgroup. We show that the simple group SL(2n…
We develop the structure theory of symplectic Lie groups based on the study of their isotropic normal subgroups. The article consists of three main parts. In the first part we show that every symplectic Lie group admits a sequence of subsequent symplectic reductions to a unique irreducible symplectic Lie group. The sec…
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.
Study Einstein Lie groups and geodesic orbit manifolds, finding some are not geodesic orbit.
problem Characterizing Einstein Lie groups and geodesic orbit manifolds.
method Characterizing GimesK-invariant geodesic orbit metrics on Lie groups G for regular subgroups K. result Extensive classes of compact simple Einstein Lie groups are not geodesic orbit manifolds.
Study homogeneous Lorentzian manifolds under reductive Lie groups, reducing descriptions to semisimple groups.
problem Characterize homogeneous Lorentzian manifolds under reductive Lie groups.
method Analyze manifolds M=G/L for connected reductive Lie groups G and reductive subgroup L; focus on totally reducible isotropy representations. result Homogeneous Lorentzian manifolds reduce to semisimple Lie groups, and are reductive.
The Greenberg-Shalom hypothesis connects subgroup properties to lattice structures in Lie groups.
problem Understanding subgroup properties in Lie groups and their implications.
method Analyzing infinite discrete subgroups of semisimple Lie groups and their commensurators.
result An infinite discrete subgroup of a semisimple Lie group with a dense commensurator is a lattice in a product of some factors.
No exact G₂-structures on compact Lie group quotients.
problem Existence of exact G₂-structures on compact quotients of Lie groups.
method Analyzing compact quotients of seven-dimensional Lie groups by co-compact discrete subgroups.
result Compact quotients of seven-dimensional Lie groups by co-compact discrete subgroups do not admit exact G₂-structures induced by left-invariant ones.
Study conformal foliations on Lie groups, finding new families and harmonic morphisms.
problem Classifying conformal foliations on Lie groups with minimal leaves.
method Analyzing left-invariant foliations generated by specific subgroups.
result New multi-dimensional families of Lie groups with conformal foliations.
New symplectic groups defined for Lie subgroups of algebras.
problem Defining new symplectic groups for Lie subgroups of algebras.
method Introducing symplectic group Sp2(G,σ) for Lie subgroups G of associative algebras A with anti-involution σ. result New realizations of spin groups as Sp2(G,σ) for suitable subgroups G. The paper finds dense subgroups in certain Lie groups.
problem Finding dense subgroups in Lie groups.
method Constructing dense surface subgroups in specific Lie groups.
result Uniform lattices contain infinitely many dense Hitchin representations.
We study infinite covolume discrete subgroups of higher rank semisimple Lie groups, motivated by understanding basic properties of Anosov subgroups from various viewpoints (geometric, coarse geometric and dynamical). The class of Anosov subgroups constitutes a natural generalization of convex cocompact subgroups of ran…
New subgroup found in Lie groups with unusual properties.
problem Finding discrete subgroups with specific properties in Lie groups.
method Constructing a specific subgroup of a higher rank Lie group.
result Found a new subgroup that is dense, discrete, non-lattice, and non-tempered.
Completed realizations of automorphisms and subgroups in exceptional Lie group E8.
problem Classifying automorphisms and subgroups of the exceptional Lie group E8. method Detailed analysis of automorphisms and subgroups induced by specific transformations.
result Explicit forms and structures of automorphisms and subgroups in E8. The paper encourages Kleinian group thinking for higher rank Lie groups.
problem No specific problem stated; encouraging new thinking.
method Discussion of Kleinian group ideas applied to higher rank Lie groups.
result Encouragement to think about higher rank Lie groups using Kleinian group theory.
The theorems of M. Ratner, describing the finite ergodic invariant measures and the orbit closures for unipotent flows on homogeneous spaces of Lie groups, are extended for actions of subgroups generated by unipotent elements. More precisely: Let G be a Lie group (not necessarily connected) and Gamma a closed subgroup …
Constructs flows on quotients of Lie groups for Anosov subgroups.
problem Understanding dynamics on quotients of Lie groups by Anosov subgroups.
method Utilizes geometric structures and Lie group theory to construct and analyze flows.
result Establishes that all refraction flows arise from this construction.
We describe explicitly Lie superalgebra isomorphisms between the Lie superalgebras of first-order superdifferential operators on supermanifolds, showing in particular that any such isomorphism induces a diffeomorphism of the supermanifolds. We also prove that the group of automorphisms of such a Lie superalgebra is a s…
Study of logarithms in SVD-closed subgroups of unitary group.
problem Understanding logarithms in SVD-closed subgroups of unitary groups.
method Analysis of generalized principal logarithms and minimizing geodesics.
result Set of generalized principal logarithms is a disjoint union of diffeomorphic subsets.
Generalizes Klein-Maskit theorem to free products of Anosov subgroups.
problem Proving a generalization of Klein-Maskit theorem for free products of Anosov subgroups.
method Analyzing Anosov subgroups in free products of semisimple Lie groups.
result The group generated by free products of Anosov subgroups is again Anosov and isomorphic to the free product.
Classifies Zariski closures of positive representations in Lie groups.
problem Classifying Zariski closures of positive representations in Lie groups.
method Classifies the Lie algebra of the Zariski closure of a discrete subgroup with specific properties.
result Obtains a new proof of Guichard's classification of Zariski closures of Hitchin representations.
We propose a sliding surface for systems on the Lie group SO(3)×R3 . The sliding surface is shown to be a Lie subgroup. The reduced-order dynamics along the sliding subgroup have an almost globally asymptotically stable equilibrium. The sliding surface is used to design a sliding-mode controller for t…