Nilpotent quandles have simple characterizations and are closely related to nilpotency.
problem Characterizing and understanding nilpotent quandles.
method Characterization of generating sets, Hopf property, construction of free nilpotent quandles, simple presentations of associated groups.
result Nilpotent quandles have the Hopf property and are equivalent to reduced peripheral systems.
Nilpotent Lie algebras obtained from ordered sets and quivers are algebraic Ricci solitons.
problem Constructing nilpotent Lie algebras as algebraic Ricci solitons
method Using transitively and antisymmetrically ordered sets (TAOSs) and incidence algebras
result Nilpotent Lie algebras with arbitrarily high degrees of nilpotency are algebraic Ricci solitons
New method constructs nilpotent Lie algebras from quivers.
problem Constructing nilpotent Lie algebras from quivers.
method Using paths within quivers to construct nilpotent Lie algebras.
result Constructs a broad family of Ricci soliton nilmanifolds.
Study knot invariants using automorphism groups of free nilpotent groups.
problem Developing knot invariants using automorphism groups.
method Nilpotently p-localization of knot groups and automorphism groups of free nilpotent groups. result Maps from outer automorphism groups yield knot invariants.
Nilpotent Sasakian manifolds are Heisenberg nilmanifolds.
problem Characterizing Sasakian manifolds with nilpotent fundamental groups.
method Proved diffeomorphism to Heisenberg nilmanifolds.
result Compact aspherical Sasakian manifolds with nilpotent fundamental groups are Heisenberg nilmanifolds.
We exhibit pseudo Riemannian manifolds which are Szabó nilpotent of arbitrary order, or which are Osserman nilpotent of arbitrary order, or which are Ivanov-Petrova nilpotent of order 3.
Classifies complex structures on specific nilpotent Lie algebras.
problem Classifying complex structures on nilpotent Lie algebras with minimal center.
method Classification through algebraic and geometric methods.
result Space of complex structures on specific Lie algebras up to isomorphism.
We compute the characteristic varieties and the Alexander polynomial of a finitely generated nilpotent group. We show that the first characteristic variety may be used to detect nilpotence. We use the Alexander polynomial to deduce that the only torsion-free, finitely generated nilpotent groups with positive deficiency…
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 limits of adjoint orbits for Lie groups, describing nilpotent orbits.
problem Understanding limits of adjoint orbits for Lie groups.
method Systematic and topological study of limits of continuous families of adjoint orbits for non-compact simple Lie groups.
result Explicit description of nilpotent orbits in terms of Richardson orbits for hyperbolic semisimple elements.
Completes classification of G2-structures on specific nilpotent Lie groups.
problem Classifying seven-dimensional nilpotent Lie groups with purely coclosed G2-structures.
method Analyzing nilpotent Lie groups of various steps and dimensions.
result Classification of indecomposable 5- and 6-step nilpotent Lie groups.
New findings on complex structures in nilpotent Lie algebras and pseudo-Kähler geometry.
problem Classifying complex structures on nilpotent Lie algebras.
method Analysis of Lie algebras in dimensions eight and higher, identifying structures with pseudo-Kähler metrics.
result Identification of pseudo-Kähler nilmanifolds with invariant complex structures, providing counterexamples and topological restrictions.
The paper studies gradings on nilpotent Lie algebras linked to smooth algebraic varieties.
problem Understanding gradings on nilpotent Lie algebras associated with algebraic varieties.
method Analyzing lattice structures in nilpotent Lie groups and their fundamental groups.
result Conditions for a lattice to be the fundamental group of a smooth complex algebraic variety.
This paper completes the classification of certain nilpotent Lie groups with specific geometric structures.
problem Classifying nilpotent Lie groups with purely coclosed G2-structures.
method Analyzing seven-dimensional nilpotent Lie groups of various steps.
result Classification of indecomposable 5- and 6-step nilpotent Lie groups with these structures.
Criterion for nilpotent Lie groups to have nilsolitons.
problem Existence of nilsolitons in nilpotent Lie groups.
method Algebraic criterion for nilpotent Lie algebras, proving necessary and sufficient condition for nilsolitons.
result Criterion provides a necessary and sufficient condition for nilpotent Lie groups to admit nilsolitons.
We study complex product structures on nilpotent Lie algebras, establishing some of their main properties, and then we restrict ourselves to 6 dimensions, obtaining the classification of 6-dimensional nilpotent Lie algebras admitting such structures. We prove that any complex structure which forms part of a complex pro…
Study proves equality of LS-category and cohomological dimension for specific group homomorphisms.
problem Proving equality of LS-category and cohomological dimension for specific group homomorphisms.
method Analyzing epimorphisms and homomorphisms between specific types of almost nilpotent and virtually nilpotent groups.
result Equality of LS-category and cohomological dimension for specified group homomorphisms.
Study nilpotent Higgs bundles and Calabi-Yau moduli metrics.
problem Understand nilpotent Higgs bundles and their metrics.
method Algebraic inequality for nilpotent matrices, geometric applications.
result Sharp upper bound of holomorphic sectional curvatures on Calabi-Yau moduli.
This paper classifies strongly nilpotent special multi-flags and their Goursat counterparts.
problem Local classification of strongly nilpotent special multi-flags and Goursat distributions.
method Study of special multi-flags in homogeneous case, focusing on their weights.
result Strongly nilpotent germs of multiflags from different singularity classes are pairwise inequivalent.
Study on pseudo-Hermitian quadratic nilpotent Lie algebras with methods and classifications.
problem Characterizing and classifying pseudo-Hermitian quadratic nilpotent Lie algebras.
method Construction methods and double extension by planes.
result Complete classification of nilpotent quadratic Lie algebras and pseudo-Hermitian metrics up to dimension 8.
This article provides a complete description of the differential Gerstenhaber algebras of all nilpotent complex structures on any real six-dimensional nilpotent algebra. As an application, we classify all pseudo-Kählerian complex structures on six-dimensional nilpotent algebras such that the differential Gerstenhaber a…
Extends Floquet-Bloch theory to nilpotent groups for geometric applications.
problem Asymptotic problems on nilpotent covers of negatively curved manifolds.
method Generalized Floquet-Bloch theory using Malcev completions.
result Branching formula relating finite and infinite-dimensional representations.
Researchers determine Dehn functions of specific nilpotent groups.
problem Understanding the Dehn functions of central products of nilpotent groups.
method Analyzing families of filiform and Lie groups to determine Dehn functions.
result Confirms conjecture and provides evidence for lower Dehn functions in central products.
Gromov proposed an averaged version of the Dehn function and claimed that in many cases it should be subasymptotic to the Dehn function. Using results on random walks in nilpotent groups, we confirm this claim for most nilpotent groups. In particular, if a nilpotent group satisfies the isoperimetric inequality $δ(l)<Cl…
We study the question of the existence of left-invariant Sasaki contact structures on the seven-dimensional nilpotent Lie groups. It is shown that the only Lie group allowing Sasaki structure with a positive definite metric tensor is the Heisenberg group. We find a complete list of the 22 classes of seven-dimensional n…
Classifies nilpotent Lie groups with specific G2-structures.
problem Identifying 7D nilpotent Lie groups with purely coclosed G2-structures. method Examined all 7D nilpotent Lie algebras, classified them, and verified the existence or non-existence of G2-structures. result Provided a complete classification of 7D nilpotent Lie groups with purely coclosed G2-structures. Study torsion-free nilpotent fundamental groups of smooth varieties up to rank 7.
problem Characterize fundamental groups of smooth quasi-projective varieties.
method Analyzes fundamental groups of smooth quasi-projective varieties using topological and Lie group theory.
result Determine fundamental groups for smooth quasi-projective varieties up to rank 7.
A near-identity nilpotent pseudogroup of order m >= 1 is a family f_1, ..., f_n: (-1,1) -> R of C^2 functions for which: |f_i - id|_{C^1} < epsilon for some small positive real number epsilon < 1/10^{m+1} and commutators of the functions f_i of order at least m equal the identity. We present a classification of near-id…
A nonpolycyclic nilpotent-by-cyclic group Gamma can be expressed as the HNN extension of a finitely-generated nilpotent group N. The first main result is that quasi-isometric nilpotent-by-cyclic groups are HNN extensions of quasi-isometric nilpotent groups. The nonsurjective injection defining such an extension induces…
The study characterizes G₂-structures on 2-step nilpotent Lie groups.
problem Characterizing G₂-structures on 2-step nilpotent Lie groups.
method Analyzing left-invariant purely coclosed G₂-structures on 7-dimensional 2-step nilpotent Lie groups.
result Criteria for Riemannian metrics induced by these structures and determination of isomorphism classes.
In this paper we attempt to give a systematic account on privileged coordinates and the nilpotent approximation of Carnot manifolds. By a Carnot manifold it is meant a manifold with a distinguished filtration of subbundles of the tangent bundle which is compatible with the Lie bracket of vector fields. This paper lies …
New definition of Rumin complex for nilpotent Lie groups.
problem No new problem introduced.
method Alternative definition of Rumin complex on nilpotent Lie groups.
result Direct application of ℓq,p cohomology results to all nilpotent Lie groups. The paper studies nilpotent structures in oriented neutral vector bundles and neutral hyperKähler structures.
problem Nilpotent structures in oriented neutral vector bundles and their relation to neutral hyperKähler structures.
method Defined H-nilpotent structures for Lie subgroups of SO(2n,2n) related to neutral hyperKähler structures. result Existence of complex and paracomplex structures forming neutral hyperKähler structures if and only if there exists an H-nilpotent structure. The study defines a canonical nilpotent structure for certain collapsed manifolds.
problem Understanding the structure of collapsed Riemannian manifolds.
method Analyzes the nilpotent structure of manifolds with bounded Ricci curvature and Reifenberg local covering geometry.
result A canonical nilpotent structure can be defined and uniquely determined over regular limit spaces.
The study classifies nilpotent Lie foliations with cohomological obstructions.
problem Understanding rigidity of nilpotent Lie foliations under solvable deformations.
method Development of a cohomological framework and algebraic criterion for rigidity.
result Established a necessary and sufficient algebraic criterion for rigidity in generalized Heisenberg groups.
The paper characterizes and examines nilpotent complex structures on stratified Lie algebras.
problem Characterizing and understanding nilpotent complex structures on stratified Lie algebras.
method Introduced a new descending series pj to prove a new characterization of nilpotent complex structures and examined whether these structures preserve the strata. result Found that there exists a J-invariant stratification on a step 2 nilpotent Lie algebra with a complex structure. The paper establishes analogs of Stallings' theorem for group homomorphisms and their nilpotent quotients.
problem Understanding the structure of fundamental groups of geometric objects.
method Develops analogs of Stallings' theorem for group homomorphisms and their nilpotent quotients.
result Derives applications including non-isomorphic number fields and hyperbolic manifolds with isomorphic universal nilpotent quotients.
Holomorphic actions on complex spaces for nilpotent groups.
problem Understanding polynomial actions on complex spaces for nilpotent groups.
method Explicit construction of biholomorphisms by polynomial maps.
result Simply connected nilpotent Lie groups are biholomorphic to Cn. Plante-Thurston proved that every nilpotent subgroup of $\Diff^2(S^1)$ is abelian. One of our main results is a sharp converse: $\Diff^1(S^1)$ contains every finitely-generated, torsion-free nilpotent group.
We associate a two-step nilpotent Lie algebra to an arbitrary Schreier graph. We then use properties of the Schreier graph to determine necessary and sufficient conditions for this Lie algebra to extend to a three-step nilpotent Lie algebra. As an application, if we start with pairs of non-isomorphic Schreier graphs co…
Study para-complex structures on specific Lie groups, finding explicit forms and properties.
problem Characterizing para-Kähler structures on six-dimensional nilpotent Lie groups.
method Examined left-invariant para-complex structures on six-dimensional nilpotent Lie groups, obtained explicit expressions and investigated curvature properties.
result Para-complex structures are nilpotent and para-Kähler metrics are Ricci-flat.
We illustrate an algorithm to classify nice nilpotent Lie algebras of dimension n up to a suitable notion of equivalence; applying the algorithm, we obtain complete listings for n≤9. On every nilpotent Lie algebra of dimension ≤7, we determine the number of inequivalent nice bases, which can be 0, 1, o…
The study constructs families of nilpotent Lie groups with geodesic orbit metrics.
problem Finding geodesic orbit metrics on nilpotent Lie groups.
method Construction of continuous families of nilpotent Lie groups.
result Continuous families of non-isomorphic nilpotent Lie groups with geodesic orbit metrics.
Study conformal Killing forms on specific nilpotent Lie groups.
problem Characterize conformal Killing forms on 2-step nilpotent Lie groups.
method Analyzing left-invariant forms on simply connected groups, proving properties of forms based on center dimension.
result Only specific forms exist under certain conditions.
In this work we study the problem of existence of symplectic structures on free nilpotent Lie algebras. Necessary and sufficient conditions are given for even dimensional ones. The one dimensional central extension for odd dimensional free nilpotent Lie algebras is also considered.
New representation for braid groups and surface braid groups, extending Lawrence-Krammer-Bigelow.
problem Constructing representations for braid groups and surface braid groups.
method Pro-nilpotent tower of representations, starting with the original LKB representation.
result 3-variable enrichment of the Lawrence-Krammer-Bigelow representation.
New examples of Lie algebras with ad-invariant metrics found.
problem Finding ad-invariant metrics on nonnice nilpotent Lie algebras.
method Introducing single extension method to construct Lie algebras with ad-invariant metrics.
result Explicit examples of nonnice nilpotent Lie algebras with ad-invariant metrics for dimensions > 10 and steps > 2.
A homogeneous nilpotent Lie group has a scaling automorphism determined by a grading of its Lie algebra. Many proofs of upper bounds for the Dehn function of such a group depend on being able to fill curves with discs compatible with this grading; the area of such discs changes predictably under the scaling automorphis…