The paper extends spectral results to non-abelian groups acting on compact Riemannian manifolds.
problem Determining potential functions from spectral data for non-abelian group actions.
method Generalized Legendrian relations and spectral invariants.
result Potential functions are determined by the equivariant spectrum for certain Schrödinger operators.
New proof shows abelian Cantor groups can act on spaces.
problem Understanding actions of Cantor groups on metric spaces.
method Examined actions of abelian Cantor groups on metric spaces.
result Cantor groups can be abelian for n>1 in space actions.
We construct homogeneous flat pseudo-Riemannian manifolds with non-abelian fundamental group. In the compact case, all homogeneous flat pseudo-Riemannian manifolds are complete and have abelian linear holonomy group. To the contrary, we show that there do exist non-compact and non-complete examples, where the linear ho…
New examples show non-abelian fundamental groups for positive Ricci curvature manifolds.
problem Constructing manifolds with positive Ricci curvature and non-abelian fundamental groups.
method Constructing specific 9-dimensional manifolds with positive Ricci curvature and non-uniformly virtually abelian fundamental groups.
result Examples of manifolds with positive Ricci curvature and non-uniformly virtually abelian fundamental groups.
Study learns convolution operators on compact Abelian groups using regularization.
problem Learning convolution operators on compact Abelian groups.
method Regularization-based approach with ridge regression estimator.
result Characterizes the accuracy of the estimator in terms of finite sample bounds.
The paper characterizes convex co-compact groups with one-dimensional boundary faces.
problem Characterizing convex co-compact groups with specific boundary properties.
method Proving relative hyperbolicity and using coarse Hilbert dimension.
result Convex co-compact groups with one-dimensional boundary faces are relatively hyperbolic.
We characterise the virtually abelian groups which are fundamental groups of compact Kähler manifolds and of smooth projective varieties. We show that a virtually abelian group is Kähler if and only if it is projective. In particular, this allows to describe the Kähler condition for such groups in terms of integral sym…
Harmonic almost complex structures on specific Lie groups and solvmanifolds identified.
problem Characterizing harmonic almost complex structures on almost abelian Lie groups and solvmanifolds.
method Adapted Gray-Hervella classification to almost abelian Lie groups, characterized harmonic structures.
result Examples of harmonic almost complex structures in different Gray-Hervella classes on compact almost abelian solvmanifolds.
Study on G2-manifolds shows automorphism group is abelian.
problem Understanding automorphism groups of G2-manifolds.
method Analyzing compact 7-manifolds with closed non-parallel G2-structures.
result The identity component of automorphism group is abelian with dimension bounded.
The study of holonomy groups in flat solvmanifolds, proving finite abelian groups can be holonomy groups and describing dimensions.
problem Understanding the holonomy groups of flat solvmanifolds.
method Elementary proof and construction of examples for various dimensions.
result Finite abelian groups can be holonomy groups of flat solvmanifolds, and specific dimensions and holonomy groups are described.
Abelian subgroups in certain spaces are simple.
problem Characterizing Abelian subgroups in specific geometric spaces.
method Analyzing the asymptotic norm of Abelian subgroups.
result Each Abelian subgroup is isomorphic to Zk. Study of complex and Hermitian structures on specific Lie groups.
problem Classifying Lie groups with specific geometric structures.
method Analysis of left-invariant structures on almost abelian Lie groups.
result Classification of six-dimensional generalized Kähler almost abelian Lie groups.
Classifies complex symplectic structures on Lie algebras with large abelian ideals.
problem Classifying complex symplectic structures on Lie algebras with large abelian ideals.
method Two constructions of complex symplectic structures on Lie algebras with large abelian ideals, considering compact quotients of Lie groups.
result Complete classification of complex symplectic structures on almost abelian Lie algebras.
Lie groups with bi-invariant distance are products of abelian and compact groups.
problem Characterizing Lie groups with bi-invariant distances.
method Analyzing the structure of Lie groups and introducing a Finsler norm.
result The sectional curvature of bi-invariant distances is non-negative and vanishes only for abelian subalgebras.
Ancient solutions on bundles are found for non-abelian groups.
problem Finding ancient solutions on bundles with non-abelian structural groups.
method Generalized ancient solutions of Ricci flow on mSO(3) bundles to RP3 fibre bundles over quaternionic Kähler manifolds. result Ancient solutions of Type I, κ-noncollapsed, and positive Ricci curvature on bundles.
The paper defines angle structures on 3-manifolds using abelian groups.
problem Finding conditions for labelings of tetrahedra in 3-manifolds.
method Non-trivial conditions on labelings of tetrahedra in a triangulated 3-manifold using an abelian group.
result Angle structures can be used to define a non-trivial condition on labelings of tetrahedra.
Classifies Spin(7) structures on compact 8-manifolds with abelian fundamental group.
problem Classifying Spin(7) structures on compact 8-manifolds.
method Obstruction theory applied to Spin(7) structures on compact 8-manifolds with abelian fundamental group.
result Compact Riemannian 8-manifolds with holonomy Spin(7) have exactly two Spin(7) structures extending the induced G2 structure on the boundary.
Compact actions on manifolds with specific eigenvalue conditions.
problem Actions of semidirect products on compact manifolds with eigenvalue constraints.
method Analyzing the induced action on cohomology and using it to generalize results.
result Existence of neighborhoods of trivial actions with abelian properties.
The paper studies a flow on complex Lie groups, showing convergence to solitons.
problem The study of curvature flows on complex Lie groups.
method Positive Hermitian curvature flow on left-invariant metrics.
result The flow converges to solitons in both nilpotent and almost-abelian cases.
The note confirms a conjecture for specific Lie groups.
problem The conjecture about constant holomorphic sectional curvature in non-Kähler geometry.
method Compact quotients of Lie groups with specific properties.
result The conjecture is confirmed for almost abelian Lie algebras and those with certain abelian ideals.
Characterizes hypercomplex Lie groups and their solvmanifolds.
problem Understanding hypercomplex structures on Lie groups and their solvmanifolds.
method Characterization of almost abelian Lie groups with hypercomplex structures, analysis of Obata and Bismut connections, classification of hypercomplex Lie groups, and construction of solvmanifolds.
result Classification of hypercomplex almost abelian Lie groups in dimension 8 and properties of their solvmanifolds.
Study shows compact mapping class groups of infinite type surfaces are never perfect.
problem Characterizing the perfection of mapping class groups of infinite type surfaces.
method Analyzing the closure of compactly supported mapping class groups and Torelli groups, examining their abelianizations.
result The abelianization of the closure of compactly supported mapping class groups contains uncountable direct sums of rationals.
Analytic torsion defined for non-compact Lie groups and discrete subgroups.
problem Defining and calculating analytic torsion for non-compact Lie groups and their discrete subgroups.
method Localised analytic torsion and relative analytic torsion defined for Lie groups of type I, using representations and discrete subgroups.
result Relative analytic torsion of (G,Γ) coincides with Lott L2 analytic torsion of a covering space. Study shows virtually abelian subgroups have commensurable counterparts in mapping class groups.
problem Understanding virtually abelian subgroups in mapping class groups.
method Proving commensurability and normalizer relationships for virtually abelian subgroups.
result Upper bounds for geometric dimension of mapping class groups for abelian subgroups of bounded rank.
We describe a generalization of GKM theory for actions of arbitrary compact connected Lie groups. To an action satisfying the non-abelian GKM conditions we attach a graph encoding the structure of the non-abelian 1-skeleton, i.e., the subspace of points with isotopy rank at most one less than the rank of the acting gro…
Develops Lie algebraic approach for compact complex homogeneous manifolds.
problem Proves important results on compact complex homogeneous manifolds.
method Uses standard results in Lie theory to associate a canonical abelian Lie algebra with a given integrable complex structure.
result Provides a new method of associating a canonical abelian Lie algebra with a given integrable complex structure.
Study balanced Hermitian structures on almost abelian Lie algebras, classifying six-dimensional cases.
problem Classify balanced Hermitian structures on almost abelian Lie algebras.
method Classify six-dimensional almost abelian Lie algebras with balanced structures, investigate flow of balanced metrics and anomaly flow.
result Prove conjecture for compact almost abelian solvmanifolds with left-invariant complex structures.
The paper proves a Whitehead theorem for fine shape spaces.
problem Proving a Whitehead theorem for fine shape spaces.
method Using Steenrod-Sitnikov homotopy groups and ind-groups.
result Fine shape morphisms are equivalences if they induce isomorphisms on π_i.
Simply connected indefinite homogeneous spaces are compact and have specific Lie algebra structures.
problem Characterizing simply connected indefinite homogeneous spaces of finite volume.
method Analyzing Lie algebras with abelian solvable radical and symmetric bilinear form.
result Simply connected indefinite homogeneous spaces are compact and have specific Lie algebra structures.
Researchers determine the rational abelianization of a subgroup of mapping class groups.
problem Understanding the structure of the Chillingworth subgroup of mapping class groups.
method Using Johnson homomorphism and Casson-Morita homomorphism, they compute the abelianization and order of related Euler classes.
result They find the rational abelianization of the Chillingworth subgroup as a full mapping class group module.
v2: An additional assumption was added in Theorem 4.8. In order to show that a connected abelian group is admissible on the site of locally compact spaces we must in addition assume that it is locally topologically divisible. This condition is used in the proof of Lemma 4.62.
This paper resolves equivariant K-theory for abelian actions.
problem Understanding equivariant K-theory for abelian group actions.
method Using iterated spaces and twisted deRham forms, the paper describes equivariant K-theory in terms of bundles over the base.
result A direct proof of the equivariant Atiyah-Hirzebruch isomorphism is provided.
Let M be a smooth compact connected oriented manifold of dimension at least two endowed with a volume form. Assuming certain conditions on the fundamental group π1(M) we construct quasi-isometric embeddings of either free Abelian or direct products of non-Abelian free groups into the group of volume preserving diffe…
We study the cohomology properties of the singular foliation $\F$ determined by an action Φ:G×M→M where the abelian Lie group G preserves a riemannian metric on the compact manifold M. More precisely, we prove that the basic intersection cohomology $\lau{\IH}{*}{\per{p}}{\mf}$ is finite dimensiona…
The main results of this article provide asymptotics at infinity of the Green's functions near and at the spectral gap edges for "generic" periodic second-order elliptic operators on noncompact Riemannian co-compact coverings with abelian deck groups. Previously, analogous results have been known for the case of $\math…
Global group laws connect equivariant bordism rings to formal group laws.
problem Establishing connections between equivariant bordism rings and formal group laws.
method Global homotopy theory framework; proving isomorphisms and universal properties.
result Equivariant bordism rings are isomorphic to Lazard rings for abelian Lie groups.
Study Picard groups of curves with symmetry, focusing on abelian groups and hyperelliptic curves.
problem Understanding the Picard groups of moduli spaces of curves with symmetry.
method Theory of symmetric mapping class groups, finitely generated Picard groups computation.
result Finitely generated Picard groups for moduli spaces of curves with abelian automorphisms.
We study HKT structures on nilpotent Lie groups and on associated nilmanifolds. We exhibit three weak HKT structures on R8 which are homogeneous with respect to extensions of Heisenberg type Lie groups. The corresponding hypercomplex structures are of a special kind, called abelian. We prove that on any 2-step nilp…
Study locally conformally balanced metrics on specific Lie algebras.
problem Characterize and classify locally conformally balanced metrics on almost abelian Lie algebras.
method Characterizations and classifications based on specific properties of Lie algebras.
result Classification of six-dimensional almost abelian Lie algebras with locally conformally balanced metrics.
Let G be a complex reductive algebraic group (not necessarily connected), let K be a maximal compact subgroup, and let A be a finitely generated Abelian group. We prove that the conjugation orbit space Hom(A,K)/K is a strong deformation retract of the GIT quotient space Hom(A,G)//G. As a corollary, we determine necessa…
Let K be a compact Lie group. We compute the abelianization of the Lie algebra of equivariant vector fields on a smooth K-manifold X. We also compute the abelianization of the Lie algebra of strata preserving smooth vector fields on the quotient X/K.
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.
A homology cylinder over a compact manifold is a homology cobordism between two copies of the manifold together with a boundary parametrization. We study abelian quotients of the homology cobordism group of homology cylinders. For homology cylinders over general surfaces, it was shown by Cha, Friedl and Kim that their …
Classifies geodesic orbit spaces with abelian isotropy subgroups.
problem Characterizing and classifying geodesic orbit spaces with specific isotropy subgroups.
method Simplified study of geodesic orbit metrics on G/S by reducing to submanifolds and generalized flag manifolds, using properties of root systems.
result Geodesic orbit spaces of the form (G/S,g) are naturally reductive.
Study lattices in specific Lie groups for geometric structures.
problem Existence of lattices in Lie groups with certain geometric structures.
method Analyzing left invariant locally conformal Kähler or symplectic structures.
result Existence of lattices only in dimension 4 for Kähler structures, and in any even dimension for symplectic structures.
Holomorphic curves found in compact quotients of SL(2,C).
problem Proving the existence of holomorphic curves in compact quotients of SL(2,C).
method Non-Abelian Hodge correspondence, WKB analysis, and Morgan-Shalen compactification.
result Every compact quotient of SL(2,C) contains a holomorphic curve of genus at least two.
We define the C^*-action on moduli spaces of reductive representations of fundamental groups of quasi-compact Kaehler manifolds by solving Hermitian-Yang-Mills equation. As applications in algebraic geometry we show a non-abelian Hodge (p,q)-type theorem for families of quasi-projective manifolds. We also prove that an…
Chern-Simons theory on a closed contact three-manifold is studied when the Lie group for gauge transformations is compact, connected and abelian. A rigorous definition of an abelian Chern-Simons partition function is derived using the Faddeev-Popov gauge fixing method. A symplectic abelian Chern-Simons partition functi…