Study on pure virtual braids, their formality, and related ranks.
problem Formality and ranks of pure virtual braid groups.
method Investigation of resonance varieties, lower central series ranks, Chen ranks, and Alexander-type invariants.
result Complete answer to 1-formality question for pure virtual braid groups.
The paper explores various braid-like groups and their properties.
problem Investigating the pure braid groups and their relatives.
method Examining resonance varieties, lower central series ranks, Chen ranks, residual and formality properties.
result Discussed natural homomorphisms and methods to distinguish braid-like groups.
Study shows shorter words for group elements in surface groups and RAAGs.
problem Understanding the structure of surface groups and RAAGs through word lengths.
method Proved lower bounds on shortest words representing nontrivial elements in their lower central series.
result Found that the shortest words are shorter for surface groups and RAAGs.
Study on virtual knot groups and their lower central series properties.
problem Understanding the structure of virtual knot groups and their lower central series.
method Analyzing groups of virtual knots with small crossings, proving properties of lower central series and decompositions.
result Existence of virtual knots with long lower central series and residually nilpotent groups.
Study on when the lower central series stops for various groups, including braid groups.
problem Understanding when the lower central series stops for different groups.
method Various techniques applied to braid groups and related groups.
result Complete computation of the lower central series for most groups studied.
Study shows fundamental group and lower central series torsion are not determined by intersection lattices of real line arrangements.
problem Determining fundamental groups and lower central series quotients of real line arrangements by their intersection lattices.
method Computer-assisted proofs and deducing from known results.
result Fundamental groups and lower central series quotients are not combinatorially determined.
We determine the lower central series and corresponding residual properties for braid groups and pure braid groups of orientable surfaces.
Study lower central and derived series of braid and pure braid groups on compact surfaces.
problem Determine residual properties of braid and pure braid groups on compact surfaces.
method Analyzing semi-direct products and calculating lower and derived series explicitly.
result Explicit calculations and estimates for residual properties of braid groups on various compact surfaces.
We determine the lower central and derived series of the n-string braid groups B_n(RP^2) of the real projective plane. We are motivated in part by the study of Fadell-Neuwirth short exact sequences, but the problem is interesting in its own right. For n=1,2, B_n(RP^2) is finite and its lower central and derived series …
Extends Milnor's invariants to 3-manifolds, solving an open problem.
problem Extract transfinite Milnor invariants for 3-manifolds.
method Develops a theory of transfinite invariants for 3-manifold groups.
result Realizes nontrivial values of transfinite Milnor invariants.
Study surjections between braid groups on surfaces, focusing on lower central series.
problem Determine conditions for surjections between braid groups on surfaces.
method Utilizes properties of lower central series and combinatorial methods.
result Identifies specific values of m and n for which surjections exist between braid groups.
We give various estimates of the minimal number of self-intersections of a nontrivial element of the kth term of the lower central series and derived series of the fundamental group of a surface. As an application, we obtain a new topological proof of the fact that free groups and fundamental groups of closed surfaces …
We prove that groups that are mod-p-homology equivalent are isomorphic modulo any term of their derived p-series, in precise analogy to Stallings' 1963 result for the lower-central p-series. Similarly spaces that are mod-p-homology equivalent have fundamental groups that are isomorphic modulo any term of their p-derive…
In 1964, John Stallings established an important relationship between the low-dimensional homology of a group and its lower central series. We establish a similar relationship between the low-dimensional homology of a group and its derived series. We also define a torsion-free-solvable completion of a group that is ana…
Let F_n be the free group on n generators. Define IA_n to be group of automorphisms of F_n that act trivially on first homology. The Johnson homomorphism in this setting is a map from IA_n to its abelianization. The first goal of this paper is to determine how much this map contributes to the second rational cohomology…
Study on braid groups' congruence subgroups and their crystallographic quotients.
problem Understanding congruence subgroups and crystallographic quotients of braid groups.
method Investigation of lower central series of congruence braid groups related to B3. result Quotients of congruence braid groups are almost crystallographic.
Study on Milnor fibrations of arrangements with trivial algebraic monodromy.
problem Explicit formulas for Milnor fiber Betti numbers in complex hyperplane arrangements.
method Analysis of cohomology jump loci and lower central series quotients of π1(F).
result Found arrangements with same Betti numbers but different fundamental groups.
Study shows lower central subgroups of a subgroup don't contain those of the free group.
problem Whether lower central subgroups of a subgroup contain those of the free group.
method Analyzes the relationship between lower central subgroups of a free group and its subgroup.
result Lower central subgroups of a subgroup do not contain those of the free group if the subgroup does not normally generate the free group.
We consider exact sequences and lower central series of surface braid groups and we explain how they can prove to be useful for obtaining representations for surface braid groups. In particular, using a completely algebraic framework, we describe the notion of extension of a representation introduced and studied recent…
Study of groups for virtual trefoil and Kishino knots.
problem Characterize groups associated with virtual trefoil and Kishino knots.
method Defined and investigated groups G1,r(L), G2(L), and G3(L) for virtual trefoil, found their structures, proved non-isomorphism, and constructed invariants. result Proved that G3 distinguishes the Kishino knot from the trivial knot. Johnson filtrations of mapping class groups are finitely generated.
problem Finiteness of Johnson filtrations in mapping class groups.
method Proved linear stable range for lower central series and Johnson filtrations of Torelli subgroups.
result Every term of the filtrations is finitely generated.
New curves share invariant up to any fixed order.
problem Finite-order rondle invariants and lower central series.
method Formulation and construction of infinite sequences of curves.
result Infinitely many curves share invariants up to any fixed order.
The study proves properties of subgroups in automorphism groups and mapping class groups.
problem Finiteness properties of subgroups in specific groups.
method Analysis of lower central series and subgroup properties.
result Subgroups containing specific terms of the lower central series have finitely generated abelianizations.
Proves finite generation of Johnson filtrations terms in mapping class groups and automorphism groups.
problem Finite generation of Johnson filtrations terms in mapping class groups and automorphism groups.
method Proves finite generation of terms in lower central series and Johnson filtrations of Torelli subgroups.
result Proves every term of the lower central series and Johnson filtrations of Torelli subgroups are finitely generated in a stable range.
Discrete commensurators of certain subgroups of PSL2(R) proven.
problem Proving discreteness of commensurators of specific subgroups of PSL2(R).
method Analyzing commensurators of terms of the lower central series or derived series of finite index normal subgroups of PSL2(Z).
result The commensurator of a specific term of the lower central series or derived series of a subgroup is discrete.
Study loop ensembles on graphs, linking group theory and topology.
problem Understanding loop homotopy classes and homologies on graphs.
method Determined distributions of loop homotopy classes and homologies using the lower central series of the fundamental group.
result Distributions of loop homotopy classes and homologies defined by the lower central series of the fundamental group.
Graph cohomology solves symplectic problems in surface mapping groups.
problem Compute the symplectic decomposition of Torelli group and its cohomology.
method Graph cohomology and ideas from graph cohomology.
result Effective computation of the symplectic decomposition of the quadratic dual of the lower central series of the Torelli group.
The McCool group, denoted PΣn, is the group of pure symmetric automorphisms of a free group of rank n. The cohomology algebra H∗(PΣn,Q) was determined by Jensen, McCammond and Meier. We prove that H∗(PΣn,Q) is a non-Koszul algebra for n≥4, which answers a question of Cohen and Pr…
The paper constructs finite generating sets for complex algebraic structures.
problem Finite generation of specific algebraic structures.
method Explicit construction of finite generating sets for γ2IAn and γ2Inb. result Explicit finite generating sets for γ2IAn and almost explicit for γ2Inb. Study various series of groups and their Lie algebras in split extensions.
problem Understanding different series of groups and their Lie algebras in split extensions.
method General construction of N-series, semi-direct products, and monodromy action.
result Generalize the well-known theorem of Falk-Randell to other versions of the LCS.
The paper proves representation stability for Torelli groups and their filtrations.
problem Representation stability of Torelli groups and their filtrations.
method Uniform representation stability using rational VICQ-modules and SIQ-modules. result The quotients of the lower central series of Torelli subgroups are uniformly representation stable.
Paper extends Stanford's equivalence to virtual knots and proves equivalence to Goussarov-Polyak-Viro's n-equivalence.
problem Characterizing finite type invariants in virtual knot theory.
method Using the lower central series of the pure virtual braid group to define an L_n-equivalence.
result L_n-equivalence on virtual string links is equivalent to Goussarov-Polyak-Viro's n-equivalence.
``What aspects of a group are unchanged, or stable, under homology equivalences''? The model theorem in this regard is the 1963 result of J. Stallings that the lower central series is preserved under any integral homological equivalence of groups. Various other theorems of this nature have since appeared. Stallings him…
We analyze the degree-two part of the Torelli group's associated graded.
problem Understanding the structure of the degree-two part of the Torelli group's associated graded.
method We use algebraic topology and group theory to analyze the structure of the degree-two part of the Torelli group's associated graded.
result The abelian group Γ2I/Γ3I is torsion-free and described as a lattice in a rational vector space. New risk-dependent centrality measures assess node importance in financial networks.
problem Understanding how external risk levels affect network node importance.
method Developed risk-dependent centrality measures based on SI model of epidemics.
result Observed ranking interlacement phenomenon where nodes can swap positions due to external risk changes.
If G and H are finitely generated, residually nilpotent metabelian groups, H is termed para-G if there is a homomorphism of G into H which induces an isomorphism between the corresponding terms of their lower central quotient groups. We prove that this is an equivalence relation. It is a much coarser relation than isom…
Let Gamma_k be the lower central series of a surface group Gamma of a compact surface S with one boundary component. A simple question to ponder is whether a mapping class of S can be determined to be pseudo-Anosov given only the data of its action on Gamma/Gamma_k for some k. In this paper, to each mapping class f whi…
We give new information about the relationship between the low-dimensional homology of a group and its derived series. This yields information about how the low-dimensional homology of a topological space constrains its fundamental group. Applications are given to detecting when a set of elements of a group generates a…
The paper refines triple linking numbers and connects them to surface systems.
problem Understanding the relationship between triple linking numbers and surface systems.
method Analyzing the homeomorphic surface systems and using group theory properties.
result Conditions for homeomorphic surface systems are equivalent to refined triple linking numbers.
We show that the Vassiliev invariants of a knot K, are obstructions to finding a regular Seifert surface, S, whose complement looks "simple" (e.g. like the complement of a disc) to the lower central series of its fundamental group.
Delta finite-type invariants are defined analogously to finite-type invariants, using delta moves instead of crossing changes. We show that they are closely related to the lower central series of the commutator subgroup of the pure braid group.
Our aim is to determine the lower central series (LCS) and derived series (DS) for the braid groups of the sphere and of the finitely-punctured sphere. We show that for all n (resp. all n\geq 5), the LCS (resp. DS) of the n-string braid group B\_n(S^2) is constant from the commutator subgroup onwards, and that Γ\_2(B\_…
Introduces arithmetic analogues of Orr invariants and spaces for absolute Galois groups.
problem Understanding arithmetic properties of absolute Galois groups through analogies with mapping class groups.
method Introduces arithmetic pro-ℓ Orr invariants and spaces, and investigates their properties and relations. result Determines the rank of the pro-ℓ Orr space as a Zℓ-module. Study non-standard bi-orders on punctured torus bundles, matching standard ones in key subgroups.
problem Investigate non-standard bi-orders on punctured torus bundles.
method Analyze various bi-orderings and compare them to standard ones formed by the lower central series.
result For every bi-ordering, the largest and second largest proper convex subgroups match those of a standard bi-ordering. Third largest subgroup matches if it exists.
PowerGossip compresses model differences for decentralized deep learning with low-rank linear compressors.
problem Communication bottleneck in decentralized deep learning models.
method Low-rank linear compressors applied on model differences using power iteration steps.
result Asymptotically independent of network and compression, faster convergence, and comparable performance to tuned compression algorithms.
Let X and Y be finite-type CW-complexes (X connected, Y simply connected), such that the rational cohomology ring of Y is a k-rescaling of the rational cohomology ring of X. Assume H^*(X,Q) is a Koszul algebra. Then, the homotopy Lie algebra pi_*(Omega Y) tensor Q equals, up to k-rescaling, the graded rational Lie alge…
Estimate arrival times in random recursive trees using iterated Jordan centralities.
problem Estimate arrival times in random recursive trees.
method Pointwise approach using iterated Jordan centralities.
result Tail bounds for relative estimation error.
This is a survey of some recent developments in the study of complements of line arrangements in the complex plane. We investigate the fundamental groups and finite covers of those complements, focusing on homological and enumerative aspects. The unifying framework for this study is the stratification of the character …