New dynamics derived from Lie groups using 2nd order tangent groups.
problem Deriving dynamics on complex Lie groups.
method Using double cross product groups and 2nd order Euler-Lagrange equations.
result 2nd order Lagrangian dynamics on double cross product groups derived.
Proves HNN extensions of nilpotent groups are left-orderable, constructs non-left-orderable examples.
problem Characterizing left-orderability in HNN extensions of groups.
method Analyzes HNN extensions of torsion-free nilpotent groups and left-orderable groups.
result Constructs examples of non-left-orderable HNN extensions of left-orderable groups.
Study of group orderability using tensor and exterior squares.
problem Circular orderability of groups and torsion elements.
method Analysis of tensor and exterior squares of groups.
result Circularly orderable groups have left-orderable tensor and exterior squares under certain conditions.
Introduces representations of surface groups into Lie groups preserving order.
problem Representations of surface groups into Lie groups that preserve the group order.
method Relates order preserving representations to weakly maximal representations and uses geometric and causal structure.
result Order preserving representations into Lie groups of Hermitian type are faithful with discrete image and form a closed set.
New criterion for bi-ordering free groups under automorphisms.
problem Bi-ordering free groups under automorphisms.
method New criterion for bi-ordering free groups.
result Fundamental group of the magic manifold is bi-orderable.
Book explores topology and orderable groups, with applications in knot theory and 3-manifolds.
problem Interplay between topology and orderable groups, focusing on fundamental groups and knot groups.
method Algebraic and topological methods, including space of orderings.
result Explicit orderings of important groups in topology and applications to knot theory.
It is well known that a countable group admits a left-invariant total order if and only if it acts faithfully on R by orientation preserving homeomorphisms. Such group actions are special cases of group actions on simply connected 1-manifolds, or equivalently, actions on oriented order trees. We characterize a class of…
Bi-orderable groups from left-orderable ones, showing non-profinite properties.
problem Non-profinite properties in bi-orderability and generalized torsion elements.
method Using fiber products to construct bi-orderable groups, showing non-profinite properties.
result Bi-orderability and generalized torsion elements are not profinite properties.
New criterion links circular-orderability to left-orderability.
problem Understanding when circularly-orderable groups are left-orderable.
method Introducing a criterion based on the product of a group with integers.
result Groups are left-orderable if their product with integers is circularly-orderable.
Generates mapping class groups with specific order elements.
problem Generating mapping class groups with elements of fixed finite order.
method Proves generation of specific order elements in mapping class groups and related groups.
result Can generate mapping class groups with 3 elements of order k and 4 elements of order 5 for sufficiently large genus. The paper explores circular orderability in 3-manifold groups, related to the L-space conjecture.
problem Circular orderability of 3-manifold groups and its relation to the L-space conjecture.
method Investigation of finite cyclic covers and Dehn surgeries to establish circular orderability.
result Circularly orderable fundamental groups of compact, connected, P^2-irreducible 3-manifolds are characterized.
New conditions for circular orderability of direct products, linking to left-orderability of groups.
problem Conditions for circular orderability of direct products GimesZ/nZ. method Cohomological conditions and characterizations for left-orderability.
result New characterization for left-orderability of fundamental groups of rational homology 3-spheres.
Study builds non-bi-orderable groups without generalized torsion.
problem Constructing non-bi-orderable groups without generalized torsion.
method Constructing specific one-relator groups.
result Demonstrates existence of non-bi-orderable groups without generalized torsion.
Study of origamis' singularities for groups of prime-power order.
problem Classifying singularities of origamis for groups of prime-power order.
method Geometric and group-theoretic ideas used to classify strata.
result Many groups of prime-power order have only one stratum, but some do not.
Proves two knots' groups are not bi-orderable.
problem Proving non-bi-orderability of specific knots.
method Broad method applicable to many knots.
result Proves 62 and 76 knot groups are not bi-orderable. We investigate the orderability properties of fundamental groups of 3-dimensional manifolds. Many 3-manifold groups support left-invariant orderings, including all compact P^2-irreducible manifolds with positive first Betti number. For seven of the eight geometries (excluding hyperbolic) we are able to characterize whi…
We are concerned with orderable groups and particularly those with orderings invariant not only under multiplication, but also under a given automorphism or family of automorphisms. Several applications to topology are given: we prove that the fundamental groups of hyperbolic nonorientable surfaces, and the groups of c…
Paper generalizes connections between Lie groups and affine connections.
problem Exploring properties of infinitesimal groups and affine connections.
method Introducing second-order infinitesimal groups and using them to define Lie brackets and connections.
result Generalized correspondence between symmetric and non-symmetric affine connections.
Paper finds conditions for free product of circularly-ordered groups to have a circular ordering.
problem Conditions for free product of circularly-ordered groups to admit a circular ordering.
method Established a categorical framework to amalgamate left-ordered central extensions, applying work of Bludov and Glass.
result Necessary and sufficient conditions for the free product with amalgamation of circularly-ordered groups to admit a circular ordering.
Study on braids and their impact on hyperbolic 3-manifolds' orderability.
problem Investigating the relationship between braids and the orderability of hyperbolic 3-manifolds.
method Examined faithful representations of braid groups on automorphism groups of free groups, focusing on bi-orderability.
result Found that the bi-orderability of hyperbolic 3-manifolds' fundamental groups is linked to the braids used.
Study finds the order of Dehn twists in various groups.
problem Determining the order of Dehn twists in different topological groups.
method Analyzing the specific groups mentioned in the abstract.
result The order of Dehn twists was calculated in the specified groups.
Automated theorem prover proves non-orderability of groups.
problem Proving non-orderability of groups.
method Generic automated theorem prover with tools like positive cones, torsions, generalised torsions, and cofinal elements.
result Demonstrated automated proof of non-orderability of groups.
Paper finds generalized torsions in non-bi-orderable 3-manifold groups.
problem Finding non-bi-orderable 3-manifold groups.
method Analyzing fundamental groups of once punctured torus bundles.
result Identifies generalized torsions in non-bi-orderable groups.
Solves the Wiegold problem by showing free products of left-orderable groups have normal rank > 1.
problem Wiegold problem about groups of normal rank > 1
method Topological argument and intricate construction of left-orders
result Free products of nontrivial left-orderable groups have normal rank > 1
Motivated by well known results in low-dimensional topology, we introduce and study a topology on the set CO(G) of all left-invariant circular orders on a fixed countable and discrete group G. CO(G) contains as a closed subspace LO(G), the space of all left-invariant linear orders of G, as first topologized by Sikora. …
Novel higher-order group synchronization for noisy local measurements on hypergraphs.
problem Synchronizing higher-order local measurements on hyperedges to global estimates on nodes.
method Message passing algorithm for global synchronization of higher-order measurements.
result Higher-order method outperforms standard pairwise synchronization methods in certain applications.
The paper characterizes L-space 3-manifolds using quandle orderability.
problem Characterizing L-space 3-manifolds using quandle orderability.
method Using quandles and their extensions, the paper investigates orderability and circular orderability conditions.
result The n-quandle Qn(L) of a link quandle is not right circularly orderable. Proves left-orderability of mapping class groups of infinite-type surfaces.
problem Left-orderability of mapping class groups of infinite-type surfaces.
method Inductive construction of a stable Alexander system and ideal arc systems.
result Proves left-orderability using carefully chosen exhaustion by finite-type subsurfaces.
Toroidal 3-manifolds have special group structures that can be shown through specific covers.
problem Characterizing the fundamental groups of toroidal 3-manifolds.
method Proving toroidal 3-manifolds have circularly-orderable fundamental groups by showing they admit finite cyclic covers with left-orderable fundamental groups.
result Toroidal 3-manifolds have circularly-orderable fundamental groups, which can be shown through specific covers.
In this paper it is proved that the pure braided Thompson's group BF admits a bi-order, analog to the bi-order of the pure braid groups.
Uniform interpretation of group theory in manifold homeomorphisms.
problem Understanding group properties in manifold homeomorphisms.
method First order theory interpretation of second order group theory.
result Many group theory problems encoded in homeomorphism groups.
New knot groups found to be bi-orderable using pretzel knots.
problem Understanding bi-orderability of knot groups.
method Applied Mayland's technique to pretzel knots to show their commutator subgroups are residually-torsion-free nilpotent.
result Found new examples of bi-orderable knot groups for non-fibered or alternating knots.
Groups acting on bifoliated planes are left-orderable.
problem Characterizing groups acting on bifoliated planes.
method Construction of a linear order on leaf space ends and identification with boundary circle subsets.
result Groups acting on bifoliated planes are left-orderable.
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.
New research shows all intermediate subgroups of braid groups are not bi-orderable.
problem Understanding bi-orderability of braid groups and their subgroups.
method Analyzing pure and full braid groups of various surfaces.
result Intermediate subgroups of braid groups are not bi-orderable.
Study automorphism groups of quandles up to order 10.
problem Understanding automorphism groups of quandles of various orders.
method Enumerated and computed automorphism groups and displacement groups of quandles up to order 10.
result Computed automorphism groups and displacement groups for all quandles up to order 10.
We show that the Bruschlinsky group with the winding order is a homeomorphism invariant for a class of one-dimensional inverse limit spaces. In particular we show that if a presentation of an inverse limit space satisfies the Simplicity Condition, then the Bruschlinsky group with the winding order of the inverse limit …
We prove that the pure braid groups on closed, orientable surfaces are bi-orderable, and that the pure braid groups on closed, non-orientable surfaces have generalized torsion, thus they are not bi-orderable.
This paper proves bi-orderability of two-bridge link groups.
problem Bi-orderability of two-bridge link groups.
method Modified graph theoretic construction of Hirasawa and Murasugi to understand Alexander subgroups.
result Bi-orderability of a large family of two-bridge link groups.
We are concerned with mapping class groups of surfaces with nonempty boundary. We present a very natural method, due to Thurston, of finding many different left orderings of such groups. The construction involves equipping the surface with a hyperbolic structure, embedding the universal cover in the hyperbolic plane, a…
There are various results that frame left-orderability of a group as a geometric property. Indeed, the fundamental group of a 3-manifold is left-orderable whenever the first Betti number is positive; in the case that the first Betti number is zero this property is closely tied to the existence of certain nice foliation…
We provide evidence for the conjecture that the Wodzicki-Chern classes vanish for all bundles with the group Z of invertible zeroth order pseudodifferential operators as structure group. In particular, we prove this vanishing if the structure group reduces to pseudodifferential operators with leading order symbol the i…
Proposes second-order influence functions for identifying influential groups in test-time predictions.
problem Identifying influential groups in test-time predictions for black-box models.
method Second-order approximations of the effect of removing a group of training samples on model predictions.
result Improves the correlation between computed influence values and ground truth values for linear models.
A positive integer m will be called a {\it finitistic order} for an element γ of a group Γ if there exist a finite group G and a homomorphism h:Γ→G such that h(γ) has order m in G. It is shown that up to conjugacy, all but finitely many elements of a given finitely generated, torsion-free Kleinian gr…
New invariant classifies right-angled Coxeter groups, bounds thickness.
problem Classifying and bounding right-angled Coxeter groups.
method Introducing hypergraph index from defining graph, computing upper bounds.
result Hypergraph index partitions groups into quasi-isometry classes, bounds thickness.
New method detects organized eCommerce fraud by clustering orders.
problem Detecting coordinated fraud by groups of fraudsters.
method Scalable categorical clustering using agglomerative clustering and sampling.
result Groups 35-45% of fraudulent orders together, detects 26.2% of fraud with low false alarms.
Alexander polynomial detects non-bi-orderability for certain knot groups.
problem Detecting bi-orderability of knot groups.
method Alexander polynomial analysis on fibered knots.
result Alexander polynomial has at least one positive real root for bi-orderable groups.
The Dehornoy order on braid groups is derived from a cluster algebra.
problem Deriving the Dehornoy order on braid groups from a cluster algebra.
method Using a tracial state on a cluster C∗-algebra associated to a surface. result The Dehornoy order on B2g+n is recovered from the cluster algebra.