New group structure from interval homeomorphisms.
problem Understanding homeomorphisms of the interval.
method Introducing chain groups and studying their properties.
result Uncountably many isomorphism types of chain groups and subgroups.
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48 results for “chain groups”
The study proves stabilizing of ascending chains in specific groups.
problem Stabilization of ascending chains in bounded rank subgroups of 3-manifold groups.
method Reduction to hyperbolic 3-manifolds and use of geometrization.
result Ascending chains in toral relatively hyperbolic groups stabilize.
Generic groups satisfy a chain condition for subgroups.
problem Understanding subgroup structures in generic groups.
method Proving for fixed integers in generic -generator -relator groups.
result Generic groups satisfy the Ascending Chain Condition for -generated subgroups.
No hyperbolic group can have an infinite chain of free subgroups of fixed rank.
problem Infinite ascending chains of free subgroups in hyperbolic groups.
method Proof by contradiction and properties of hyperbolic groups.
result Hyperbolic groups do not contain strictly ascending chains of free quasiconvex subgroups of constant rank.
The study provides bounds for geodesic diameter in Euclidean space.
problem Finding bounds for geodesic diameter in Euclidean space.
method Develops a geometric approach using locally rectifiable chains and complete normed commutative group bundles.
result Provides a new method for calculating geodesic diameter bounds.
Study on CR structures on 3D Lie groups, focusing on equivalence and closed chains.
problem Characterizing CR structures on 3D Lie groups and their chains.
method Analyzing left-invariant CR structures on 3D Lie groups and their equivalence.
result All chains on G are closed if and only if G is CR equivalent to specific spherical CR structures.
Maps quandle homology to relative group homology.
problem Understanding the relationship between quandle and group homology.
method Introducing a chain map and constructing quandle cocycles.
result Relates quandle homology to relative group homology through triangulations.
The paper proves an ascending chain condition for subgroups in hyperbolic and graph 3-manifolds.
problem Proving an ascending chain condition for subgroups in specific types of 3-manifolds.
method Uses profinite techniques and geometric proofs for hyperbolic and graph manifolds.
result Established the ascending chain condition for free subgroups of constant rank in closed hyperbolic and graph 3-manifolds.
A theorem simplifies mass-minimizing flat chains' regularity.
problem Understanding the regularity of mass-minimizing flat chains.
method Simple condition for fundamental regularity principle.
result Fundamental regularity principle holds for mass-minimizing chains.
We prove (without using Federer's structure theorem) that a finite-mass flat chain over any coefficient group is rectifiable if and only if almost all of its 0-dimensional slices are rectifiable. This implies that every flat chain of finite mass and finite size is rectifiable. It also leads to a simple necessary and su…
Proves quaternionic analog of Cartan's theorem and counts arithmetic chains.
problem Understanding transformations of quaternionic hyperbolic spaces.
method Analyzes chain-preserving transformations and arithmetic chains in quaternionic Heisenberg group.
result Proves analog of Cartan's theorem and provides counting and equidistribution results.
We define an algebraic group comprising symmetric chain complexes which captures the first two stages of the Cochran-Orr-Teichner solvable filtration of the knot concordance group in a single invariant. To achieve this we impose additional structure on each chain complex which puts extra control on the fundamental grou…
We define an algebraic group comprising symmetric chain complexes which captures the first two stages of the Cochran-Orr-Teichner solvable filtration of the knot concordance group in a single obstruction. To achieve this we impose additional structure on each chain complex which puts extra control on the fundamental gr…
This paper presents the theory of non-smooth Lie group actions on chains of Banach manifolds. The rigorous functional analytic spaces are given to deal with quotients of such actions. A hydrodynamical example is studied in detail.
Study on stable commutator length in RAAGs and Coxeter groups, proving spectral gaps and hardness results.
problem Understanding stable commutator length in right-angled Artin and Coxeter groups.
method Established spectral gaps, determined sizes up to constants, and related to graph properties.
result Found that stable commutator length can be arbitrarily close to zero in some groups, contrasting uniform gaps.
Khovanov homology is simplified for plumbing operations.
problem Understanding Khovanov homology for plumbing operations.
method Isomorphic decompositions of chain groups for link diagrams and Kauffman states.
result Explicit construction of enhancements representing nonzero Khovanov homology.
Extends Jones' construction to Thompson's group F and link homology.
problem Link homology and Thompson's group F.
method Asymptotic construction of -tangles in Thompson's group F.
result Asymptotically faithful association of -tangles with elements of Thompson's group F.
Study shows torsion homology growth vanishes for certain free-by-cyclic groups.
problem Understanding torsion homology growth in free-by-cyclic groups.
method Analyzing polynomially growing monodromy and showing vanishing homology torsion.
result Integral torsion equals -torsion for these groups, verifying a conjecture.
Proves uniqueness of holomorphic quilts on surfaces.
problem Computing boundary maps of immersed Lagrangian Floer chain groups.
method Constructs holomorphic quilts from bigons on surfaces.
result Uniqueness of holomorphic quilts provides a combinatorial method for computing boundary maps.
New manifolds help understand group actions on complex chains.
problem Understanding compact Lie group actions on Morse and Floer chains.
method Introduced new manifolds called forest biassociahedra and bimultiplihedra.
result Derived algebraic structures like bialgebras and bimodules.
Study two-generated subgroups of homeomorphisms with specific chain supports and find uncountably many isomorphism types.
problem Characterize two-generated subgroups of homeomorphisms with specific chain supports and their square roots.
method Analyze subgroups of with supports forming a chain of two intervals, and examine their square roots.
result Uncover uncountably many isomorphism types of subgroups with specific chain supports and their square roots.
The equivalence of Heegaard Floer homology and embedded contact homology III: from hat to plusmath.GT
Given a closed oriented 3-manifold M, we establish an isomorphism between the Heegaard Floer homology group HF^+(-M) and the embedded contact homology group ECH(M). Starting from an open book decomposition (S,h) of M, we construct a chain map Φ^+ from a Heegaard Floer chain complex associated to (S,h) to an embedded co…
New model captures time series dependence across and within blocks.
problem Complex multivariate time series dependence structures.
method Time series Gaussian chain graph models with directed and undirected edges.
result Consistent recovery of time series chain graph structure.
Improved sampling for network community detection.
problem Inefficient sampling from network partition posterior distributions.
method Merge-split Markov chain Monte Carlo for efficient sampling.
result Significantly improved mixing time and correct sampling.
On homology torsion growthmath.GT
New results on homology torsion growth for various groups.
problem Understanding the growth of higher torsion homologies for arithmetic lattices and other groups.
method Quantitative homotopical method called effective rebuilding, constructing small classifying spaces of finite index subgroups.
result Strong asymptotic bounds for the torsion growth in principal congruence subgroups.
We propose a simple method to produce quandle cocycles from group cocycles, as a modification of Inoue-Kabaya chain map. We further show that, in respect to "universal central extended quandles", the chain map induces an isomorphism between their third homologies. For example, all Mochizuki's quandle 3-cocycles are sho…
Alternative proof of uniform boundary condition using geometric Følner arguments.
problem Uniform boundary condition in normed chain complexes.
method Geometric Følner arguments on the chain level.
result Integral refinements of the uniform boundary condition.
We develop a theory of chain complex double-cobordism for chain complexes equipped with Poincaré duality. The resulting double-cobordism groups are a refinement of Ranicki's torsion algebraic -groups for localisations of a commutative ring with involution. The refinement is analogous to the difference between metabo…
Study isotropic embeddings of Lie group orbits and their application to magnetic geodesic flows.
problem Embedding coadjoint orbits and their equivalence to magnetic geodesic flows.
method Review and initiate study of isotropic and Lagrangian embeddings for and cases, then apply to magnetic geodesic flows.
result Equivalence between magnetic geodesic flows and certain spin chains.
We define the twisted Blanchfield pairing of a symmetric triad of chain complexes over a group ring Z[G], together with a unitary representation of G over an Ore domain with involution. We prove that the pairing is sesquilinear, and we prove that it is hermitian and nonsingular under certain extra conditions. A twisted…
This study explores factors and enablers of compliance risk in Vietnamese seafood supply chain.
problem Compliance risk in Vietnamese seafood supply chain.
method Comprehensive literature review and expert interviews.
result Three main critical factor groups influencing compliance risk identified.
Constructs manifolds from quantum codes with novel geometric properties.
problem Creating manifolds with specific geometric constraints.
method Reverse engineering manifolds from quantum code chain complexes.
result First examples of power law systolic freedom.
New mapping class group actions on Hochschild complexes for modular categories.
problem Understanding actions of mapping class groups on Hochschild complexes of modular categories.
method Construction of a symmetric monoidal functor with excision property.
result Homotopy coherent projective action of mapping class groups on Hochschild complexes.
We use the heat flow on the loop space of a closed Riemannian manifold to construct an algebraic chain complex. The chain groups are generated by perturbed closed geodesics. The boundary operator is defined in the spirit of Floer theory by counting, modulo time shift, heat flow trajectories that converge asymptotically…
GAttNHP predicts future events in temporal knowledge graphs by encoding long-range dependencies and handling mutual excitation.
problem Forecasting future events in temporal knowledge graphs due to long-range dependencies, mutual excitation, and heavy-tailed inter-arrival times.
method GAttNHP uses a self-attention encoder, semantic soft-grouping, and NCQ regression to address these issues.
result GAttNHP improves entity and time prediction on six benchmark TKG datasets compared to state-of-the-art baselines.
This paper shows a category equivalence between Lie group representations and Lie algebra representations.
problem Understanding the relationship between Lie group representations and Lie algebra representations.
method Establishes an equivalence between two categories of modules and representations.
result The equivalence between DG-algebra of singular chains on Lie groups and DG-Lie algebra representations.
We use the Yang-Mills gradient flow on the space of connections over a closed Riemann surface to construct a Morse-Bott chain complex. The chain groups are generated by Yang-Mills connections. The boundary operator is defined by counting the elements of appropriately defined moduli spaces of Yang-Mills gradient flow li…
New instanton invariants for rational homology spheres defined and shown to be functorial.
problem Defining and proving invariance of instanton homology groups for rational homology spheres.
method Novel suspended flow category technique to handle obstructed cobordisms and prove wall-crossing formula.
result Instanton invariant conjecturally equals Casson-Walker invariant for rational homology spheres.
The Waldhausen construction of Mayer-Vietoris splittings of chain complexes over an injective generalized free product of group rings is extended to a combinatorial construction of Seifert-van Kampen splittings of CW complexes with fundamental group an injective generalized free product.
We adapt to an infinite dimensional ambient space E.R. Reifenberg's epiperimetric inequality and a quantitative version of D. Preiss' second moments computations to establish that the set of regular points of an almost mass minimizing rectifiable chain in is dense in its support, whenever the group of …
A formula connects two algebraic structures derived from a category.
problem Connecting two algebraic structures derived from a category.
method Using differential graded modular functors and Calabi-Yau structures.
result The action of a specific mapping class group element transforms one algebraic structure into another.
Model learns collective and individual dynamics in time series data.
problem Lack of models capturing system-level collective behavior in individual time series.
method Hierarchical switching-state model with latent system-level and entity-level Markov chains.
result Model improves interpretability and forecasting accuracy compared to larger models.
The paper computes torsion invariants for groups acting on complexes.
problem Computing torsion invariants for groups acting on complexes.
method Analyzes residually finite groups acting cocompactly on contractible complexes with specific stabilizers.
result Torsion limits to the torsion of the boundary subcomplex, independent of the chain of subgroups.
In this paper, we examine mapping class group relations of some symplectic manifolds. For each and , we show that the -dimensional Weinstein domain , determined by the degree homogeneous polynomial , has a Boothby-Wang type boundary …
New shifting chain map enhances quandle invariants for links.
problem Enhancing quandle invariants for links and surfaces.
method Introducing a shifting chain map σ and its pull-back σ# to transform cocycles.
result Shifting chain map σ# transforms 2-cocycles to 3-cocycles, enhancing invariants.
Polar Homologymath.AG
For complex projective manifolds we introduce polar homology groups, which are holomorphic analogues of the homology groups in topology. The polar k-chains are subvarieties of complex dimension k with meromorphic forms on them, while the boundary operator is defined by taking the polar divisor and the Poincare residue …
New topological realization of Kontsevich graph complex for large dimensions.
problem Understanding the rational homotopy groups of Diff partial(D2k).
method Construction of a chain map from Kontsevich graph complex to rational singular chain complex.
result New elements in rational homotopy groups of BDiff partial(D2k) determined by cycles in graph complex.
The algebraic -groups $L_*(\A,X)$ are defined for an additive category $\A$ with chain duality and a -set , and identified with the generalized homology groups $H_*(X;\LL_{\bullet}(\A))$ of with coefficients in the algebraic -spectrum $\LL_{\bullet}(\A)$. Previously such groups had only been defined for…