New proof and description of commutator subgroups for free and surface groups.
problem Understanding commutator subgroups of free and surface groups.
method Geometric proof and representation-theoretic description.
result New free generating sets and structure descriptions for commutator subgroups.
Each element of the commutator subgroup of a group can be represented as a product of commutators. The minimal number of factors in such a product is called the commutator length of the element. The commutator length of a group is defined as the supremum of commutator lengths of elements of its commutator subgroup. We …
Study on infinite-type surfaces shows stable commutator length is continuous and defines open subgroups.
problem Understanding stable commutator length on infinite-type surfaces.
method Analyzing mapping class groups of infinite-type surfaces, showing continuity and openness of commutator subgroups.
result Stable commutator length defines a continuous function on commutator subgroups of infinite-type mapping class 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.
For any group, there is a natural (pseudo-)norm on the vector space B1 of real (group) 1-boundaries, called the stable commutator length norm. This norm is closely related to, and can be thought of as a relative version of, the Gromov (pseudo)-norm on (ordinary) homology. We show that for a free group, the unit ball of…
An arbitrary homomorphism between groups is nonincreasing for stable commutator length, and there are infinitely many (injective) homomorphisms between free groups which strictly decrease the stable commutator length of some elements. However, we show in this paper that a random homomorphism between free groups is almo…
In this paper, we investigate some applications of commutator subgroups to homotopy groups and geometric groups. In particular, we show that the intersection subgroups of some canonical subgroups in certain link groups modulo their symmetric commutator subgroups are isomorphic to the (higher) homotopy groups. This give…
Let WBn be the welded (or loop) braid group on n strands, n≥3. We investigate commutator subgroup of WBn. We prove that the commutator subgroup WBn′ is finitely generated and Hopfian. We show that WBn′ is perfect if and only if n≥5. We also compute finite presentation for FWBn′, the commuta…
Study of commutator subgroups and crystallographic quotients of virtual groups.
problem Investigate commutator subgroups and crystallographic quotients of virtual groups.
method Derived explicit finite presentations and proved crystallographic properties.
result Explicit finite presentations of commutator subgroups and crystallographic quotients.
We solved a conjecture about braid group quotients being alternating groups.
problem Understanding the smallest non-trivial quotients of braid group commutator subgroups.
method Proved the conjecture about alternating groups as quotients, showed minimal quotient maps.
result Proved conjecture about braid group quotients being alternating groups.
Doodles link to commutator identities in a 2-sphere.
problem Understanding commutator identities in free groups via doodles.
method Analyzing doodles with proper noose systems and establishing bijections.
result A bijection between doodles and commutator identities.
We give new upper bounds on the stable commutator lengths of Dehn twists in mapping class groups and new lower bounds on the stable commutator lengths of Dehn twists in hyperelliptic mapping class groups. In particular, we show that the stable commutator lengths of Dehn twists about a nonseparating and a separating cur…
We show that stable commutator length is rational on free products of free Abelian groups amalgamated over Zk, a class of groups containing the fundamental groups of all torus knot complements. We consider a geometric model for these groups and parameterize all surfaces with specified boundary mapping to th…
Paper proves non-compact inaudibility of symmetry and commutativity.
problem Proving inaudibility of symmetry and commutativity in non-compact settings.
method Using isospectral pairs of generalized Heisenberg groups.
result Proved inaudibility of weak symmetry and commutativity.
We show that the mapping class group of any closed connected orientable surface of genus at least five is generated by only two commutators, and if the genus is three or four, by three commutators.
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.
Study relative commutants in von Neumann algebras using contraction notions.
problem Understanding relative commutants in group and tracial crossed product von Neumann algebras.
method Introducing contraction notions to study relative commutants.
result Results applied to negatively curved groups and SL(d, Z).
New neural networks for non-commutative data.
problem No existing neural networks suitable for non-commutative data.
method Developed compact matrix quantum group equivariant neural networks.
result Characterized weight matrices for easy compact matrix quantum groups.
Commutes Pansu pullback with spectral complexes in Carnot groups.
problem Understanding the relationship between Pansu pullback and spectral complexes in Carnot groups.
method Proving commutativity between Pansu pullback and differentials in spectral complexes.
result Commutes Pansu pullback with spectral complexes in Carnot groups.
Abstract: Proves non-abelian group of equivariant concordance.
problem Equivariant concordance group non-abelian
method Infinite family of nontrivial commutators
result Equivariant concordance group is not abelian
Study mixed commutator lengths in wreath products and their relation to general ranks.
problem Understanding mixed commutator lengths in wreath products and their relation to general ranks.
method Analyzing wreath products (G,N)=(Z≀Γ,⨁ΓZ) and determining mixed commutator lengths in terms of general rank. result Mixed commutator lengths and ordinary commutator lengths coincide under certain conditions.
We give a new upper bound on the stable commutator length of Dehn twists in hyperelliptic mapping class groups, and determine the stable commutator length of some elements. We also calculate values and the defects of homogeneous quasimorphisms derived from ω-signatures, and show that they are linearly independent in th…
The paper classifies topological holonomy groups in SO(3).
problem Classifying holonomy groups in SO(3). method Analyzing and categorizing groups based on their properties.
result Different types of holonomy groups in SO(3) are identified and classified. Study on homeomorphism groups of telescoping 2-manifolds showing strong distortion.
problem Characterizing the homeomorphism group of telescoping 2-manifolds.
method Introduced telescoping 2-manifolds, studied homeomorphism groups, and used commutator subgroup properties.
result Homeomorphism group of telescoping 2-manifolds is strongly distorted.
We define a quasihomomorphism from braid groups to the concordance group of knots and examine its properties and consequences of its existence. In particular, we provide a relation between the stable four ball genus in the concordance group and the stable commutator length in braid groups, and produce examples of infin…
New quasimorphisms show stable commutator lengths are not equivalent.
problem Equivalence of stable commutator lengths in groups.
method Invariant quasimorphisms for groups acting on the circle.
result Stable commutator lengths are not bi-Lipschitzly equivalent.
We show that, in compact semisimple Lie groups and Lie algebras, any neighbourhood of the identity gets mapped, under the commutator map, to a neighbourhood of the identity.
Let G be a finitely presented group, and G' its commutator subgroup. Let C be the Cayley graph of G' with all commutators in G as generators. Then C is large scale simply connected. Furthermore, if G is a torsion-free nonelementary word-hyperbolic group, C is one-ended. Hence (in this case), the asymptotic dimension of…
For n at least 7 and n equal to 5, we give generating sets of size 2 for the commutator subgroup of the braid group on n strands. These generating sets are of the smallest possible cardinality. For n equal to 4 or 6, we give generating sets of size three. We also prove that the commutator subgroup of the braid …
New algebraic structures on manifolds generalize supergeometry concepts.
problem Developing algebraic structures for non-commutative manifolds.
method Introducing ρ-commutative manifolds, Q-manifolds, and modular classes. result Generalized modular classes for non-commutative spaces.
This paper classifies commutativity spaces for 3-manifold groups.
problem Classifying commutativity spaces for geometric 3-manifold groups.
method Using geometric realization of order complexes of cosets of abelian subgroups.
result For closed orientable geometric 3-manifolds, the commutativity space is homotopy equivalent to a wedge of circles.
Proves that emergent algebras right-distributivity implies left-distributivity.
problem Proving the implication between emergent algebra distributivity conditions.
method Analyzing families of quasigroup operations indexed by commutative groups.
result Emergent algebras right-distributive imply left-distributive.
Investigates BNSR invariants of link and knot groups, proving specific properties.
problem Characterizing finiteness properties of normal subgroups in link and knot groups.
method Analyzes BNSR invariants of link and knot groups, proving specific properties.
result Proves specific conditions for finiteness properties of link and knot groups.
Study on integrability of geodesic flows on Heisenberg group.
problem Integrability of geodesic flows on Heisenberg group.
method Investigation of two classes of normal geodesic flows associated with left-invariant sub-Riemannian metric.
result Left-left configuration is completely integrable in non-commutative sense, while left-right configuration exhibits non-commutative integrability in dimensions > 5.
Infinitesimal calculations link fundamental groups to Lie algebras.
problem Calculating logarithm maps in fundamental groups.
method Hopf invariants defined by Harrison cohomology of commutative cochains.
result Zeroth Harrison cohomology is a universal dual to Malcev Lie algebra.
Establishes a duality theorem connecting quasimorphisms and commutator lengths in group theory.
problem Connecting quasimorphisms and commutator lengths in group theory.
method Geometric interpretation and algebraic proof of (G,N)-commutator lengths. result Bi-Lipschitz equivalence of scl on [G,N] under certain conditions. For the free group Fr on r>1 generators (respectively, the free product G1∗G2 of two nontrivial finite groups G1 and G2), we obtain the asymptotic for the number of conjugacy classes of commutators in Fr (respectively, G1∗G2) with a given word length in a fixed set of free generators (respecti…
New proof shows rationality of scl for non-filling curves.
problem Understanding stable commutator length in non-filling curves.
method New proof using extremal surfaces for scl.
result Rationality of stable commutator length for non-filling curves.
We give examples of finitely presented groups containing elements with irrational (in fact, transcendental) stable commutator length, thus answering in the negative a question of M. Gromov. Our examples come from 1-dimensional dynamics, and are related to the generalized Thompson groups studied by M. Stein, I. Liousse …
Maps commuting with sub-Laplacians on Carnot groups are conformal.
problem Characterizing maps preserving sub-Laplacians on sub-Riemannian Lie groups.
method Analyzing smooth maps between sub-Riemannian Lie groups that commute with sub-Laplacians.
result Sub-Laplacian determines the sub-Riemannian structure in Carnot groups.
Let Γ be a finite index subgroup of the mapping class group MCG(Σ) of a closed orientable surface Σ, possibly with punctures. We give a precise condition (in terms of the Nielsen-Thurston decomposition) when an element g∈Γ has positive stable commutator length. In addition, we show that in these situations th…
New homeomorphism found in Klein bottle group.
problem Understanding homeomorphisms of Klein bottle.
method Using recent results on commutator length.
result Existence of homeomorphism with positive stable commutator length.
The study examines spaces of non-extendable quasimorphisms for group pairs.
problem Analyzing the space of non-extendable quasimorphisms for group pairs.
method Established a five-term exact sequence of cohomology relative to bounded subcomplex.
result Proved stable commutator length equivalent to stable mixed commutator length for certain pairs.
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.
The paper extends ternary algebra concepts using cube roots of unity.
problem Extending algebraic structures from binary to ternary multiplication.
method Introducing ternary associator, commutator, and Lie algebra at cube roots of unity.
result Derived an identity for ternary commutator based on GA(1,5). 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.
The paper characterizes crystallographic groups derived from virtual braid and twin groups.
problem Characterizing crystallographic groups from virtual braid and twin groups.
method Analyzing quotients of virtual braid and twin groups by their commutator subgroups.
result The quotients of virtual braid and twin groups by their commutator subgroups are crystallographic groups.
Let Hn denote the complex hyperbolic space of dimension n. The group U(n,1) acts as the group of isometries of Hn. In this paper we investigate when two isometries of the complex hyperbolic space commute. Along the way we determine the centralizers.